10

Chapter 2

Mathematical Language and Symbols

LEARNING OUTCOME(S):
 Compare and contrast mathematical expression and sentences.
 Understand the basic concepts of mathematics such as sets, functions and relations.
 Identify the truth value of compound propositions and perform basic operations on
logic.

TIME FRAME: 6 hours

LESSON 2.1 THE LANGUAGE OF MATHEMATICS

Students frequently have trouble understanding mathematical ideas: not necessarily because
the ideas are difficult, but because they are being presented in a foreign language which is the
language of mathematics. The language of mathematics can be learned, but requires the efforts
needed to learn any foreign language.

The language of mathematics makes it easy to express the kinds of thoughts that
mathematicians like to express. It is:
 precise (able to make very fine distinctions);
 concise (able to say things briefly);
 powerful (able to express complex thoughts with relative ease).

Every language has its vocabulary (the words), and its rules for combining these words into
complete thoughts (the sentences). Mathematics is no exception.

In English, nouns are used to name things we want to talk about (like people, places, and
things); whereas sentences are used to state complete thoughts. A typical English sentence has
at least one noun, and at least one verb. For example, consider the sentence

Jason loves mathematics.

Here, ‘Jason’ and ‘mathematics’ are nouns; ‘loves’ is a verb.

An expression is the mathematical analogue of an English noun (people, places, and things); it is
a correct arrangement of mathematical symbols used to represent a mathematical object of
interest. An expression does NOT state a complete thought; in particular, it does not make
sense to ask if an expression is true or false.

Numbers are the most common type of mathematical expression. And, numbers have lots of
different names. For example, the expressions all look different, but are all just different names
for the same number.
 11

5 2+3 10÷2 (8 ÷ 2) + 1 1+1+1+1+1

A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of

mathematical symbols that states a complete thought. It makes sense to ask about the TRUTH
of a sentence: Is it true? Is it false? Is it sometimes true/sometimes false?

Just as English sentences have verbs, so do mathematical sentences. “1 + 2 = 3” is read as “one

plus two equals three” or “one plus two is equal to three”. A complete thought is being stated,
which in this case is true. Indeed, the equal sign `=' is one of the most popular mathematical
verbs.

The symbol `+' is a connective; a connective is used to `connect' objects of a given type to get a
`compound' object of the same type. Here, the numbers 1 and 2 are `connected' to give the
new number 1 + 2. A familiar English connective for nouns is the word `and': `cat' is a noun,
`dog' is a noun, `cat and dog' is a `compound' noun.

ENGLISH MATHEMATICS
name given to an object of NOUN EXPRESSION
interest: (person, place, thing) 5 , 2 + 3 , 1/2
Examples: Carl, Manila, book
a complete thought: SENTENCE SENTENCE
The capital of Philippines is 3+4=7
Manila. 3+4=8
The Carl is reading a book.

One way to decide whether something is a sentence, or not is to read it, and ask the question:
Does it state a complete thought? If the answer is “yes”, then it's a sentence.

Consider, for example, the number 1 + 2 . Read it: “one plus two”. Have you stated a complete
thought? NO! But, if you say: 1 + 2 = 4, then you have stated a complete (false) thought.

Alternately, you can ask yourself the question:

Does it make sense to ask about the TRUTH of this object? Consider again the number “1+2”. Is
“1+2” true? Is “1+2”false? These questions don't make sense, because it doesn't make sense to
ask about the truth of an expression!
WORKSHEET 4

NAME: SCORE:
 12

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Circle the verbs in the following sentences and answer the following with TRUE or
FALSE:
a. The capital of Philippines is Davao.
b. 3 + 4 = 7

2. Classify each entry as a mathematical expression (EXP), or a mathematical sentence

(SEN). Then determine the truth value of each entry that you identified as a sentence
(SEN): (always) true (T); (always) false (F); or sometimes true/sometimes false (ST/SF).
The first two are done for you.
Classification always/Sometimes True/False
(sample) 1 + 2 EXP
(sample) 1 + 2 = 3, SEN T
a) 1
b) x+1
c) x +1 = 3
d) x-3
e) x-3=2
f) 1+2+x=x+1+2

3. Use the English noun ‘Mark’ in three sentences: one that is true, one that is false, and
one whose truth cannot be determined without additional information.

Always true:

Always false:
cannot be determined:

4. Use the mathematical expression ‘3’ in three sentences: one that is true, one that is
false, and one whose truth cannot be determined without additional information.
Always true:

Always false:
cannot be determined:
 13

LESSON 2.2 BASIC CONCEPTS OF MATHEMATICS

Language of Sets
Sets were introduced by George Cantor (1845-1918). A set must be well defined; i.e., for any
given object, it must be clear whether or not the object is an element of the set. Objects in sets
can be anything. It can be people, physical objects, numbers, signs, other sets, etc. A set is
usually designated by a capital letter. If a set contains all the chairs in a designated room, then
any chair can be determined either to be in or not in the set. If there were no chairs in the
room, the set would be called the empty {}, or null ϕ set, i.e., one containing no elements.

Roster Method (Tabular Form) is a method where the set is represented by listing all its
elements. Two distinct elements are separated by a comma, and braces {}, are used to enclose
the listed elements of a set. If A is the set of even numbers between 1 and 9, then
A={2 , 4 , 6 , 8 }.

The elements of a set may be described without actually being listed; this is the Rule method.
We indicate a set by enclosing in braces a descriptive phrase, and agreeing that those objects,
and only those, which have the described property are elements of the set. If B is the set of real
numbers that are solutions of the equation x 2=9 , then the set can be written as B= { x| x 2=9 },
which is read B is the set of all x such that {x} ^ {2} =9; hence B is the set {3 ,−3 }.

Membership in a set is indicated by the symbol ∈ and non-membership by ∉; thus, x ∈ A

means that element x is a member of the set A (read simply as x is an element of A ) and y ∉ A
means y is not an element of A.

If it is possible to list down all the members or elements of a given set, then the set is said to be
finite. Otherwise, the set is said to be infinite.

Finite Sets
1. Let A be the set of counting numbers from 1 to 7.
Then A={ 1, 2 , 3 , 4 , 5 , 6 ,7 } Tabular or Roster method
A={ x| x are counting number ¿ 1 ¿ 7 } Rule Method
5 ∈ A , 0 ∉ A , 0 .5 ∉ A
2. Let B be the set of positive divisors of 12.
Then B= {1 , 2 ,3 , 4 , 6 ,12 } Tabular or Roster method
B= { x| x are positive divisors of 12 } Rule Method
2 ∈ B ,5 ∉ B

Infinite Sets
1. Let C be the set of integers.
Then C={ … ,−3 ,−2 ,−1 , 0 , 1 ,2 , 3 , … } Tabular or Roster method
C={ x| x are integers } Rule Method
−2 ∈C ,5 ∈C , 0.5 ∉C
 14

2. Let D be the set of prime numbers.

Then D= { 2, 3 , 5 ,7 , 11 , 13 , … } Tabular or Roster method
D= { x| x are prime numbers } Rule Method
17 ∈ D , 1∉ D ,8 ∉ D

Empty Sets
1. Let E be the set of integers between 3 and 4.
Then E={ ϕ } or E={}
−2 ∉C ,0 ∉C

Subsets and Counting

If A is any set, then the number of elements of a set is called the cardinality of A
and is denoted n( A). A set is finite if it consists of a specific number of different elements;
otherwise a set is infinite. If A is infinite, then we write, n( A)=∞.

1. Let M be the set of the cities or municipalities of Bukidnon.

Then M is finite and n(M )=22.
2. Let T ={x x is any person whose first name isTeresa }.
Then T is finite.
3. Let N={1,3,5,7 , … }. Then N is infinite and
n(N )=∞ .
A set which has only one element is known as a singleton set.
A set which has no element is said to be an empty set. This is sometimes called the null or void
set and is denoted byϕ .

1. H={ x x is a solution of 2 x−3=5 }

The solution of 2 x−3=5 is 4. Here, H={4 } .
2. P={ x x is a current governor of Bukidnon }
In any particular time, Bukidnon has one and only one governor.
3. S={x x isa 100 year old student of BSU }=ϕ={}
According to statistics, S is an empty set.
4. R={x x isboth an odd∧an even number }
R=ϕ, for there exists no number x which is both an odd and an even number.

Two sets A and B are said to be equal if they have precisely the same elements. We denote this
by A=B. A set does not depend on the way in which the elements are displayed and a set
remains the same if its elements are repeated or rearranged. If the sets are not equal then we
write A ≠ B.

Equal Sets
 15

1. Let A be the set of the letters of the word “veil” and B be the set of letters in the word
“evil”. Then A=B since
A={ e , i ,l , v } and B= { e , i, l , v }
2. Let P= { x| x∣6 ,0< x< 4 } and Q= { x| x arethe first three counting numbers }. Then P=Q
since
P= {1 , 2 ,3 , } and Q= {1 , 2 ,3 , }
3. Let A={1,5,8,9 } and B={9,1,5,8 }.
Then A=B.
4. Let E={3,4,5,4 }, F={3,4,5,5 , } and G={3,4,5 }.
Then E=F=G.
5. Let Z={x x ( x−3)=0 }, X ={0,3 },Y ={0,3,3 } and W ={3,3,3 , … , 0,0,0 , …}.
Then Z=X =Y =W .

If the elements in the two sets can be put into a one-to-one correspondence, the sets are said
to be equivalent sets. In general, two finite sets A and B with the same number of elements are
equivalent sets denoted by A ↔ B or A ≅ B. Equivalent sets are not necessarily equal but equal
sets are equivalent sets.

1. Let A={∗, ¿ , @} and B={4 ,1 , 5}. Then A ≅ B.

¿ # @

4 15

2. Let O={1,3,5,7 , … } and E={2,4,6,8 , … }. Then O ≅ E.

If every element in a set A is also a member of a set B, then A is called a subset of B. We

denote this relationship by A ⊂B. In the case that A ⊂ B but A ≠ B, we say that A is a proper
subset of B.

If A ⊂ B, then B is called a superset of A, written B⊃ A.

Remarks: Let A , B∧C be any sets. Then
1. A ⊂ A.
2. ϕ ⊂ A
3. If A ⊂ B and B⊂ A , then A=B.
4. If A ⊂ B and B⊂C, then A ⊂ C.
5. The total number of subsets of a set with n elements is 2n.

Subsets
 16

1. Let A be the set of all BSU students and B be the set of all freshmen students.
Then B⊂ A.
2. If V ={a , e , i , o ,u }. Then{a , o } and {a , e , u } are subsets of V .
3. Let X ={1,3,5,7 , … } and Y ={5,15,25,35 , …}. Then Y ⊂ X.
4. Let A={1,2,3 }. Then
the proper subsets of A are: {1 },{2},{3 }, {1,2 }, {1,3 }, {2,3}
and improper subsets of A are: A and ϕ.
5. A={1,2,3 } and n( A)=3=n
number of subsets=2n=23=2 x 2 x 2=8 subsets

If sets A and B have no elements in common, then we say that A and B are disjoint sets.

1. If A={1,3 } and B={2,4,5,6 }. Then A and B are disjoint sets.

2. The set of even numbers and the set of odd numbers are disjoint sets.

In any application of set Theory, all sets under investigation are subsets of a fixed set called the
universal set. It is usually denoted by U.

1. If X ={1,2 }, Y ={2,5,3 }, X ={1,2,4 }∧W ={1,2,3,4,5 }. Then W can be considered as the

universal set of this problem.
2. If E={2,4,6 , … }∧O={1,3,5 , …}, then N={1,2,3,4 , … } can be considered as the
universal set.
3. In plane geometry, the universal set consists of all points on the plane.

In the case that the elements of a given set are set themselves, then we say that the given set is
a family of sets (set of sets or class of sets). To avoid confusion between ordinary set and family
of sets, we will use script letters { A , B ,C , … } to denote the family/class/set of sets.

If A is any set, then the family of all subsets of A is called the power set of A, and is denoted
P(A).

Power set

1. X ={{1}, ϕ ,{a , b }}
2. If A={a , b }, then the power set of A is the set:
P( A)={ϕ , {a },{b }, A }.
3. Let S be a set with n(S)=3, then n(P (S))=23 =8.

WORKSHEET 5
 17

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

Direction: Give the corresponding Roster form or Set Builder notations of the given sets in the
table below.

SET BUILDER FORM

ROSTER FORM
(Rule Method)
1. B = { x|x is a primary color }
C = { Sunday, Monday, Tuesday, Wednesday,
2.
Thursday, Friday, Saturday }
3. E = { x|x is an integers between
-5 to 3 }
4. F = { a, e, i, o, u }

5. G = {3, 6, 9, 12, 15, 18, 21, …}

6. H = { x|x is a positive divisors of 32 }

7. I = { x|x is a real number that is a
solution to the equation x 2=25 }

1. If A={ 2 , 4 , 5 }, what is the power set of A?

2. Rewrite “ B contains X ” in set notation.

3. If the power set of P ( C ) ={ ϕ , C , { 3 } , { 4 } , { 5 } , { 3 , 4 } , { 4 ,5 } , { 3 ,5 } }, what is the set C?

4. Tell which of the following are true and which are false given that
A={ 0 , 1, 2 ,3 , 4 } B= {0 ,1 , 2 }; C={ 5 , 3 , 4 } and D= { 3 , 4 , 5 }

Statement TRUE/FALSE Statement TRUE/FALSE

a. B⊂ A f. ϕ ⊂ B
b. C ∊ A g. B ≅ D
c. { 3 } ⊂ C h. C=D
d. D∧C are disjoint sets i. n ( A ) =4
e. D ⊂ C j. A is a finite set
WORKSHEET 6
 18

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Let A and B be sets defined in the following diagram. State whether each of the following is
true or false.

Ax yw z B

Statement TRUE/FALSE Statement TRUE/FALSE

a. {x , y }❑ A f. { ϕ } ⊂ B
b. z A g. B ≅ A
c. x A h. A=B
d. A∧B are disjoint sets i. n ( A ) =3
e. {w }❑B j. A is a finite set

2. What is the universal set if A={−1 ,−2 ,−3 , … } and B= { x| x are whole numbers }

3. Let A={ x|3 x=9 } and b=3. Is the statement b ∈ A true? Explain your answer.

4. If X ={{x , u }, { y , z }}, how many subsets does X contain? Find the power set of X .

Operation on Sets
 19

In any discussion the set of all elements under consideration must be specified, and it is called
the universal set. If the universal set is U, then the complement of A(written A ' )is the set of all
elements in the universal set that are not in A.

Complement
1. If the universal set is U ={1 ,2 , 3 , 4 , 5 } and A={1 , 2, 3 }, then A ' ={4 , 5} .
2. Let U ={ 1, 2 , 3 , 4 , 5 ,6 ,7 , 8 , 9 , 10 } and B= { x| x is odd∧x <10 }. Then the elements of B
are { 1 ,3 ,5 , 7 , 9 }. Thus B' ={2 , 4 , 6 , 8 ,10 }

There are three basic set operations: intersection, union, and complementation. Set
complementation is a unary operation since it involves only one set. Intersection and union of
two sets are examples of binary operation of sets.

The intersection of two sets is the set containing the elements common to the two sets and is
denoted by the symbol ∩.
A ∩ B= { x| x belongs ¿ A∧B }.

The union of two sets is the set containing all elements belonging to either one of the sets or to
both, denoted by the symbol ∪.
A ∪ B= { x| x belongs ¿ A∨¿ B∨both}.

These two operations each obey the associative law and the commutative law, and together
they obey the distributive law. The intersection of a set and its complement is the empty, or
A ∩ A ' =∅; the union of a set and its complement is the universal set, or A ∪ A '=U.

Intersection and union

if C={1 , 2 ,3 , 4 } and D={3 , 4 , 5 },
then C ∩ D={3 , 4 } and C ∪ D={1 ,2 , 3 , 4 , 5 }.

Another important binary operation involving two sets A and B is the cross product of A and B
denoted by A × B. We define this as follows:

A × B={( x , y )∨x belongs ¿ A y belongs ¿ B }.

Cross product
1. if M ={2, 3 , 5 ,7 } and N={1 ,2 , 7 }, then
M × N= { (2,1 ) , ( 2,2 ) , ( 2,7 ) , ( 3,1 ) , (3,2 ) , ( 3,7 ) , ( 5,1 ) , ( 5,2 ) , ( 5,7 ) , ( 7,1 ) , ( 7,2 ) , ( 7,7 ) }
The difference of two set A and B, written A−B, is the set of elements which belong to A but
which does not belong to B.
Difference
A−B={ x| x ∈ A∧x B }
if C={1 , 2 ,3 , 4 } and D={3 , 4 , 5 }, then C−D={1 ,2 }
 20

Relationships between sets can be illustrated using geometric figures called Venn diagrams
introduced by the English mathematician, John Venn. A rectangle is used to represent the
universal set and circle/ellipse is used to represent the ordinary sets.

U U

A A B

U U

A B A B

In figure 1, set A is a subset of the universal set U. In figure 2, sets A and B have elements in
common which is the shaded portion but neither one is a subset of the other. The shaded
portion in figure 3 is the union of sets A and B. Sets A and B in figure 4 is obviously disjoint sets.

Venn diagram

1. A group of 45 Filipino tourist went on summer vacation this year, of this number, 23
went to Boracay, 20 went to Cebu, 16 went to Palawan, 9 went to Boracay and Cebu, 7
went to Cebu and Palawan, 5 went to Boracay and Palawan, and 3 went to all these
three places.
U
a. How many of them visited Boracay only? B C
6
b. How many visited Cebu and Palawan 12 7
but not Boracay? 3
c. How many of them went to Palawan but not Cebu? 2 4
d. How many did not go to any of these three places? 4 P 7
 21

WORKSHEET 7

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Given U ={ x∨x is a counting number less than 10 }

A={ x∨x is a prime number less than 10 }
B= { x∨x is a positive divisor of 8 }

Find the elements of each set and find its cardinality.

Elements Cardinality
'
a. A =¿
b. B' =¿
c. A ∪ B=¿
d. A ∩ B=¿
e. A' ∪B' =¿
f. ( A ∪ B )' =¿
g. A' ∩B ' =¿

2. Given A={ a , b , c , d , e } and B= { a , e , i, o , u }

a. Find A × B=¿

b. Find A−B=¿

3. An insurance company classifies its policy holders U

according to age, sex, and marital status. Of 500 policy Male
holders, it was found out that: 350 were married, 240 Married
were under 25, 230 were married men, 110 were married
and under 25, 100 were men under 25, 40 were married
men under 25, and 10 were single women 25 or older.
How many policy holders were men? Under
25 Y.O.
 22

Language of Relations and Functions

The shared property of association of terms or objects is very important in the study of the
relationships between different quantities. This shared property is known as “relation”. While
relation serves as the guiding bond between quantities, a special type of relation serves as the
real world model in solving problems. This is known as a function.

Defi niti on of Relati ons and Functi ons

In many common relationships between two variables, the value of one of the variables
depends on the value of the other. One example of these relationships of two variables is a
person’s salary. The salary of an employee depends on the time he works and on the rate of the
salary. Symbolically, we can represent the salary as S, the time as t, and the rate of salary as
r. Since the salary depends on the time and rate, mathematically we have the equation S=rt.
This equation portrays that S is a function of r and t.

A relation is a mapping that assigns each of the elements of the first set to one or more
elements of the second set. This means that all the elements in the first set must have a
corresponding image in the second set, regardless of the number of its images.

A function is a rule that produces a correspondence from a set X of real numbers x to a set Y
of real numbers y, where the number y is unique for a specific value of x. Thus, a function is a
mapping that assigns to each element of the first set a unique element in the second set. As an
ordered pair, a function is a set of ordered pairs of real numbers ( x , y ) in which no two ordered
pairs have the same first number.

The illustrations below differentiate a function from a relation.

1 1 a
1 a 1 a
2 A 2
2 b b
3 b
3 c 2 c
3 b 4
d
c
Figure 1 Figure 2 Figure 3 5Figure 4

Figures 1 and 2 shows both a function and a relation, while Figure 3 shows only a
relation. Figure 4 does not show a function, not even a relation. Why?

The mapping of the elements in Figure 1 is said to be a one-to-one mapping. This kind of
mapping maps only one element in the first set to one and only one element in the second set.
Figure 2 is a many-to-one mapping. This mapping provides two or more elements in the first set
that map to only one element in the second set. Figure 3 illustrates a one-to-many mapping.
Notice that one element in the first set has at least one corresponding value in the second set.
These mappings can also define a function and a relation. A function is a one-t-one and many-
 23

to-one mapping, while a relation is a one-to-one, many-to-one and one-to-many mapping. This
further implies that all functions are relations while not all relations are functions.
The set of elements in the first group of every mapping are known as admissible values of x and
is called the domain of the function/relation while the set of elements in the second group are
the resulting values of y and is called the range of the function/relation.

Since relation or function has been referred to as ordered pairs, we can rewrite the mappings of
Figure 1-3 as follows:

A={ ( 1 , a ) , ( 2 , b ) , ( 3 ,c ) }, where the domains are { 1,2,3 } and the range are {a ,b , c }
B= { (1 , a ) , ( 2 ,a ) , ( 3 , a ) } , where the domains are { 1,2,3 } and the range is { a } alone
C={ ( 1 , a ) , (1 , b ) , ( 2 ,b ) , ( 2 , c ) }, where the domains are { 1,2 } and the range are { a , b , c }

Example 1 Determine whether the set S= { ( 1,4 ) , ( 2,3 ) , ( 3,2 ) , ( 4,3 ) , ( 5,4 ) } defines a function.
We only need to take note about this: “there should be no two first elements can be seen in the
set”. Since the first elements in the set are different from each other, set S defines a function.

Example 2 Determine whether the set T ={ ( 1,4 ) , ( 2,3 ) , ( 1,2 ) , ( 4,3 ) , ( 4,4 ) } defines a function.
The set T does not define a function since there are ordered pairs with the same first
components. These are the ordered pairs ( 1,4 ) , ( 1,2 ) , ( 4,3 ) , and ( 4,4 ).

Function Notation
Functions are usually given in terms of equations rather than ordered pairs. These equations
are expressed in special notation. To indicate that y depends on x, we write y=f ( x ) read as “ y
equals f of x”. Here, y is the image of x under the function f . Thus, f (x) is the number
associated with x.

When a function is defined by an equation in x and y, it is given implicitly. If it is possible to

solve for y in terms of x, and write y=f ( x ), the said function is given explicitly. The statement,
4−3 x
for example, 3 x+ 2 y =4 is the implicit form of the function, while the statement f ( x )= is
2
the equivalent explicit form of it.

Given the concept above, function notation provides a way of indicating the image of a
particular number. If y=f ( x )=x−4 defines a function f , then the symbol f (3) indicates the
image of 3 under the function f .

Example 3 If f ( x )=2 x−5, find f ( 2 ) , and f ( a−2 ) .

Solution What we need is only to substitute the domains into the function and simplify the
result.
(a) f ( x )=2 x−5
f ( 2 ) =2 ( 2 ) −5=4−5=−1 (-1 is the image of the domain 2)
f ( a−2 ) =2 ( a−2 )−5=2 a−4−5=2 a−9
 24

(2 a−9 is the range of the domain a−2)

Finding the Domain and Range of a Functi on
Agreement on Domains and Ranges:
If a function is defined by an equation and the domain is not indicated, then we assume that
the domain is the set of all real number replacements of the independent variable that produce
real values for the dependent variable. The range is the set of all values of the dependent
variable corresponding to these domain values. Thus, domains are set of values for which the
function is always defined.

Example 5 Find the domain and the range of the function defined by f ( x )=2 x−3.
Solution In this function, whatever value that will be assigned to x, the result will always be
obtained and defined. Thus, the domain and the range consists all real numbers.

Domain={ x|x ∈ R }
Range= { y| y ∈ R }.

Example 6 Find the domain and the range of the function defined by g ( x )=√ x−2.
Solution Square root of negative numbers would result to imaginary numbers. For g ( x ) to
be real, x−2 must greater than or equal to zero. Notice also that the result will
always be a 0 or a positive integer. That is,
x−2 ≥ 0 or x ≥ 2.

Thus, the domain and the range will be

Domain={ x|x ∈ [ 2 , ∞ )∨x ≥2 }

Range= { y| y ∈ [ 0 , ∞ )∨ y ≥ 0 }.

2x
Example 7 Find the domain and the range of the function defined by h ( x )= .
x+3
Solution The given function is a fractional function. When the denominator becomes zero,
the function becomes undefined. If we letx=−3, the function will become
meaningless. The domain will be all real numbers except−3. Notice that if we
substitute any value to the function except 1, the result will always be in the set of
real numbers. Thus,

Domain={ x|x ∈ R / x ≠−3 }

Range= { y| y ∈ R / y ≠ 2}.
 25

WORKSHEET 8

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

A. Determine whether the following sets define a function or not. Write F if it is a function and
F ' if it is not.
1. A={ ( a , d ) , ( c , d ) , ( e , f ) , ( b , f ) }
2. B= { (1,3 ) , ( 2,4 ) , ( 5,6 ) , ( 3,4 ) }
3. C={ ( 1 , a ) , (2 , b ) , ( 3 , c ) , ( 2 , a ) , ( 4 , d ) , ( 5 , c ) }
4. D= { ( a , 1 ) , ( b , 3 ) , ( b ,3 ) , ( c , 4 ) }
5. E={ ( a , 2 ) , ( b , 4 ) , ( c , 6 ) , ( b , 8 ) , ( c ,10 ) }

B. Evaluate the following functions.

x
1. Let f ( x )=4 x 2−3 x+ 6. Find 2. Let g ( x )= . Find
x −3
a. f ( 3 ) a. g ( 3 )

b. f (x+ h) b. g ( x−3 )

C. Find the domain and the range of the following functions. Write answers in set builder form.
1. f ( x )=3 x 2−2 x+ 1

2. g ( x )=2 x −1, where x ≠−2

3. h ( x )=2 x−√ x+1

4. k ( x )=√ 3 x−2
 26

3x
5. p(x) ¿
x+2

LESSON 2.3 ELEMENTARY LOGIC

An elementary statement (or atomic statement) is a sentence with a subject and a verb (and
sometimes an object) but no connectives (and, or, not, if then, if, and only if). For example,

“Sisyle is beautiful”
“Ethan has toys”
“Claire reads books”

are all atomic statements. We build up sentences, or propositions, from atomic statements
using connectives.

Propositional Equivalences
A proposition is a declarative sentence that is either true or false (but not both). For instance,
the following are propositions:

“Malaybalay is in Bukidnon” (true),

“Valencia is in Davao” (false),
“2 < 4” (true),
“4 = 7 (false)”.

However the following are not propositions:

“what is your name?” (this is a question),

“do your homework” (this is a command),
“this sentence is false” (neither true nor false),
“x is an even number” (it depends on what x represents),
“Socrates” (it is not even a sentence).

The truth or falsehood of a proposition is called its truth value.

Connectives, Truth Tables.

Connectives are used for making compound propositions. The main ones are the following (p
and q represent given propositions):

Name Representation Meaning

Negation p “not p”
Conjunction p ˄q “p and q”
Disjunction p ˅q “p or q (or both)”
Exclusive Or p⨁q “either p or q, but not both”
Implication p →q “if p then q”
 27

Biconditional p ↔q “p if and only if q”

The truth value of a compound proposition depends only on the value of its components.
Writing F for “false” and T for “true”, we can summarize the meaning of the connectives in the
following way:

p q ~p p ˄q p ˅q p⨁q p →q p ↔q
T T F T T F T T
T F F F T T F F
F T T F T T T F
F F T F F F T T

Note that ˅ represents a non-exclusive or, i.e., p ˅q is true when any of p, q is true and also
when both are true. On the other hand ⊕ represents an exclusive or, i.e., p ⨁ q is true only
when exactly one of p and q is true.

Tautology, Contradiction, Contingency.

1. A proposition is said to be a tautology if its truth value is T for any assignment of truth values
to its components. Example: The proposition p ˅ p is a tautology.

2. A proposition is said to be a contradiction if its truth value is F for any assignment of truth
values to its components. Example: The proposition p ˄ p is a contradiction.

3. A proposition that is neither a tautology nor a contradiction is called a contingency.

p p p˅ p p˄ p
T F T F
T F T F
F T T F
F T T F

tautology contradiction

Conditional Propositions.
A proposition of the form “if p then q” or “p implies q”, represented “ p →q” is called a
conditional proposition. For instance: “if John is from Malaybalay then John is from Bukidnon”.
The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or
consequent.

Note that p →q is true always except when p is true and q is false. So, the following sentences
are true: “if 2< 4 then Malybalay is in Bukidnon” (true→true), “if Valencia is in Davao then 2< 4”
(false→true), “if 4=7 then Bukidnon State University is in Davao” (false→false). However the
following one is false: “if 2< 4 then Malaybalay is in Davao” (true→false).
 28

In might seem strange that “ p →q” is considered true when p is false, regardless of the truth
value of q. This will become clearer when we study predicates such as “if x is a multiple of 4
then x is a multiple of 2”. That implication is obviously true, although for the particular case
x=3 it becomes “if 3 is a multiple of 4 then 3 is a multiple of 2”.

The proposition p↔q, read “p if and only if q”, is called biconditional. It is true precisely when p
and q have the same truth value, i.e., they are both true or both false.

Logical Equivalence.
Note that the compound propositions p →q and p ˅q have the same truth values:
p q p ˅q p →q
p
T T F T T
T F F F F
F T T T T
F F T T T

When two compound propositions have the same truth values no matter what truth value their
constituent propositions have, they are called logically equivalent. For instance p→q and
p˅q are logically equivalent, and we write it:

p →q ≡ p ˅ q

Note that that two propositions A and B are logically equivalent precisely when A ↔ B is a
tautology.

Example: De Morgan’s Laws for Logic. The following propositions are logically equivalent:
( p ˅ q)≡ p ˄ q
( p ˄ q)≡ p ˅ q

We can check it by examining their truth tables:

p q p ˅q ( p ˅ q) p˄ q p ˄q ( p ˄ q) p˅ q
p q
T T F F T F F T F F
T F F T T F F F T T
F T T F T F F F T T
F F T T F T T F T T

Example: The following propositions are logically equivalent: p↔q ≡ (p→q)˄(q→p)

Again, this can be checked with the truth tables:
p q (p→q) (q→p) (p→q)˄(q→p) p↔q
T T T T T T
 29

T F F T F F
F T T F F F
F F T T T T
Converse, Contrapositive.
The converse of a conditional proposition p →q is the proposition q → p. As we have seen, the
biconditional proposition is equivalent to the conjunction of a conditional proposition an its
converse.
p ↔q ≡( p → q)˄(q → p)
So, for instance, saying that “Alfeo is married if and only if he has a spouse” is the same as
saying “if Alfeo is married then he has a spouse” and “if he has a spouse then he is married”.

Note that the converse is not equivalent to the given conditional proposition, for instance “if
John is from Malaybalay then John is from Bukidnon” is true, but the converse “if John is from
Bukidnon then John is from Malaybalay” may be false.

The contrapositive of a conditional proposition p→q is the proposition q→ p. They

are logically equivalent. For instance the contrapositive of “if John is from Malaybalay then John
is from Bukidnon” is “if John is not from Bukidnon then John is not from Malaybalay”.

A predicate or propositional function is a statement containing variables. For instance “ x +2=7

”, “X is Filipino”, “x < y”, “p is a prime number” are predicates. The truth value of the predicate
depends on the value assigned to its variables. For instance if we replace x with 1 in the
predicate “x +2=7” we obtain “1+2=7”, which is false, but if we replace it with 5 we get “
5+2=7”, which is true. We represent a predicate by a letter followed by the variables enclosed
between parenthesis: P(x), Q(x, y), etc. An example for P(x) is a value of x for which P(x) is true.
A counterexample is a value of x for which P(x) is false. So, 5 is an example for “ x +2=7”, while
1 is a counterexample.

Each variable in a predicate is assumed to belong to a universe (or domain) of discourse, for
instance in the predicate “n is an odd integer” ’n’ represents an integer, so the universe of
discourse of n is the set of all integers. In “X is American” we may assume that X is a human
being, so in this case the universe of discourse is the set of all human beings.

Quantifiers.
Given a predicate P(x), the statement “for some x, P(x)” (or “there is some x such that p(x)”),
represented “∃ x P ( x)”, has a definite truth value, so it is a proposition in the usual sense. For
instance if P(x) is “x +2=7” with the integers as universe of discourse, then ∃ x P ( x) is true,
since there is indeed an integer, namely 5, such that P(5) is a true statement. However, if Q(x) is
“2 x=7” and the universe of discourse is still the integers, then ∃ xQ(x ) is false. On the other
hand, ∃ xQ(x ) would be true if we extend the universe of discourse to the rational numbers.
The symbol ∃ is called the existential quantifier.

Analogously, the sentence “for all x, P(x)” also “for any x, P(x)”, “for every x, P(x)”, “for each x,
P(x)”, represented “∀ x P(x )”, has a definite truth value. For instance, if P(x) is “x +2=7” and
 30

the universe of discourse is the integers, then ∀ x P(x ) is false. However if Q(x) represents “
( x +1)2 =x2 +2 x+1 ” then ∀ xQ( x) is true. The symbol ∀ is called the universal quantifier.
WORKSHEET 9

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Determine whether the statement is a proposition or not. Write “P” if it is a proposition and
“N” if it is not a proposition. Also, identify the truth value of those statements that are
propositions, write “T” if its truth value is true, “F” if it is false and “NA” if it is identified as
not a proposition.

Statement Proposition or not? Truth value

1. Let it go!
2. 5 ¿ 21
3. Ice floats on water
4. 3 is a multiple of 6
5. Flag
2. For each of the following statements, formulate an English sentence that is its negation:
Pigs can fly.
Negation:

Pigs eat grass and goat eats grass.

Negation:

3. Write the following statement in symbolic form using the following propositions:
p: she is beautiful
q: she is happily married
r: she is rich
Statement Symbolic form
6. She is happily married and wealthy but not beautiful.
7. She is not wealthy, but she is happily married and
beautiful.
8. She is neither happily married, nor wealthy, nor smart.
9. She is rich, happily married and beautiful.
10. She is poor but beautiful.
4. Construct truth tables for each of the following sentences:
(p ∧ q) ∨ ∼ (p ∨ q) (p ∨ q) → (p ∧ q)
p q p∧q p∨q (p ∨ q) → (p ∧ q) (p ∧ q) ∨ ∼ (p ∨ q)
(p ∨ q)
 31

T T
T F
F T
F F
WORKSHEET 10

NAME: SCORE:

SECTION CODE OR CLASS SCHEDULE: DATE:

1. Determine whether the statement is a tautology, contradiction, or contingency.

( p ∧q) ∨(∼ p ∨( p ∧∼ q))∨ r

p q r p ∧q p ∧∼q ∼ p ∨( p ∧∼ q ) ( p ∧q) ∨(∼ p ∨( p ∧∼ q))∨ r

p q
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

Answer:

2. Write “≅” between the two statements if they are logically equivalent, and write “≇” if they
are not.

(p → q) p∧ q
p→q q→ p
(p ↔ q) p⨁q

p q p→q p↔q p ⨁ q p∧
p q (p→q) (p ↔ q) q→
q
p
T T
T T
T F
T F
32