GE 7 MMW Module 2
GE 7 MMW Module 2
GE 7 MMW Module 2
0 10-July-2020
Combined Mathematical
English Translations
Sentences
- The sum of six and two all over four is two.
- The ratio of six plus two, and four is equal to two.
- Two is the quotient when the sum of six and two is divided by four.
- Twice the difference of twelve and four is sixteen. - The product of 2 and twelve less four
is sixteen
2
PANGASINAN STATE UNIVERSITY
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
The set composed of five vowels of the English alphabet may be named and can
Example 1 be denoted as
Sets like has a finite number of elements. It can be written using roster
{ }
method, where the elements are listed. Commas are used
between each element and a pair of braces is used to enclose the elements.
The set whose elements are all even integers may be named and can be written as
Example 2 {
. This is read as “set is the set}of all values of such that is an even integer”. Sets like has
infinite number of elements. It is written
using set-builder method. This notation is used whenever it is convenient or impossible to list all the elements
of a set; it merely describes the characterizing property of its elements in terms of symbols.
a. Set is the set whose elements are numbers greater than negative five but less than four.
b. Set is the set whose elements are numbers greater than or equal to zero.
c. Set is the set of even numbers greater than or equal to four but less than or equal to twenty.
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
Cardinality of a set is the number of elements in it. For example, the cardinal number of set
A (described above) is denoted by . Take note that in finding
for the cardinality of a set, elements that are listed more than once are
counted only once. For example the set has a cardinality of 4 and is denoted as
Equivalent Sets are sets that have the same number of elements. In other words, they have the same
cardinality. For example, set and (discussed above) are equivalent sets denoted by since
they have one element each; that is . However, set and set are not equivalent because set has five
elements and set has only four elements.
Equal sets are sets that have exactly the same elements. o If and , then
sets and are equal, denoted by . o If and , then .
The symbol is used to indicate that an element belongs to a set; while is used to indicate that an element
does not belong to a set. o Given , we say that is an element of ” or in symbols, .
The symbol is used to indicate that set is a proper subset of another set. In given two sets and
, every element of set is also an element of set , but not all elements of set are in set . Such a
relation between sets is denoted by and is read as “ is a proper subset of ”. It is important to note that the
set on the right of is the one with more elements.
On the other hand, the symbol is used to indicate that a set is not a subset of another set; meaning not all
elements of the first set are also element of the second set.
The symbol is used to indicate that equal sets are subset of one another. Suppose we are given two sets,
and . Since sets and are equal sets, we say that is a subset of
, and conversely, is a subset of . In set notation, we state and .
Power set is the set composed of all the subsets of a given set. For example, the power set of set denoted
as is .
Note that an empty set is a subset of every set and every set is a subset of itself.
Sets that have common elements are called joint sets; while those that do not have common elements are
called disjoint sets.
Your turn 3 Fill in the table below with corresponding notation of statement and evaluate if
what it states is true or false.
5
PANGASINAN STATE UNIVERSITY
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
Your turn 4
Do you notice any relation between the number of elements in a set and number of
elements in its power set; that is the number of subsets (proper subsets plus its equal
set)?What seems to be the pattern? Can you come up with a formula? Go and
investigate! Put your observations in the table.
or set
or
both
a.
The intersection of two sets and is the set of elements that belong to both sets and , and is denoted
by which reads as intersection .
If D is the set of single-syllable months, E is the set of months with letter and
Example 4 F is the set of months with four letters or less, then D = {March, May, June}
{January, February, July} and {May, June, July}.
Find , and .
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
Your turn 6
Perform the indicated operation. Use the same sets given in example 4.
a.
b.
c.
Example 6 Let
Find and .
Solution:
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
Your turn 8
a.
b.
c.
d.
The Cartesian product of two sets and is the set of all possible pairs of elements and is denoted by
(read as “the Cartesian product of and ”). Each pair of elements is called an ordered pair
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
Solution :
a. is represented by all the regions common to
circles A and C . Thus is represented by regions i and iv.
Figure 3
Solution:
a. { } c. {}
b. { } d. { }
Counting problemsoccur in many areas of applied mathematics. To solve these counting problems,
we often make use of a Venn diagram. In the next example, we can use
diagram
Venn to help us solve some
word problems.
Solution
A Venn diagram can be used to illustrate the results of the survey. We use two overlapping circles
(see Figure 4). One circle represents the set of people who like action adventures and the other represents
the set of people who like comedies. The region i where the circles intersect represents the set of people who
like both types of movies.
We start with the information that 180 people like both types of movies and write 180 in region i. See Figure 5.
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
a. Regions i and ii have a total of 695 people. So far we have accounted for 180 of these people in region i.
Thus the number of people in region ii, which is the set of people who like action adventures but do not like
comedies, is .
b. Regions i and iii have a total of 340 people. Thus the number of people in region iii, which is the set of
people who like comedies but do not like action adventures, is .
c. The number of people who do not like action adventure movies or comedies is represented by region iv.
The number of people in region iv must be the total number of people, which is 1000, less the number of
people accounted for in regions i, ii, and iii, which is 855. Thus the number of people who do not like either
type of movie is
An activities director for a cruise ship has surveyed 240 passengers. Of the 240
Your turn 10 passengers,
{ }
LEARNING POINTS
Sets like has a finite number of elements. We use roster method, which is, listing all elements of the set,
in describing
{ finite sets. }
The set whose elements are all even integers may be named and can be written as
. This is read as “set is the set of all values of such that is an even integer”. Sets
like has infinite number of elements. It is written using set-builder method. This notation is used whenever it
{ }
{ }
{ }
{ } { } { } { }
is convenient or impossible to list all the elements of a set; it merely describes the characterizing property of its
elements in terms of symbols.
The different set operations are Union, Intersection, Difference, Complementation, and Cartesian product.
LEARNING ACTIVITY 2
In exercises 1 to 3, use the roster method to write each of the given sets .
1. The set of whole numbers less than 4.
2. The set of negative integers between -5 and 7.
3. The set of integers x that satisfy
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
In exercises 16 to 17, if {even counting numbers } and {odd counting numbers}, then which of the following
are true or false .
16. (2, 3)
17. 22
In exercises 18 to 19 . Draw a Venn diagram with each of the given elements placed in the correct region.
18.
{ }
{ }
{ }
{ }
19. Hal, Marie, Rob, Armando, Joel, Juan, Melody}
{
{Marie, Armando, Melody}
{Rob, Juan, Hal}
{Hal, Marie, Rob, Joel, Juan, Melody}
Mrs. Cruz asked her 30 students who among their mother, father, or sibling will attend the quarterly
conference. Sixteen students said their mother will attend, another 16 said their father will attend, and 11 said
their siblings will attend. Five said their mother and sibling will attend, and of these, 3 said their father will also
attend. Five said only their sibling will attend and 8 said only their father will attend. How many students said
only their mother will attend? Support your answer by illustrating the Venn Diagram which represents the
given data.
Solution
a. This is not a proposition because it is not a declarative sentence.
b. This sentence is a proposition because it is a declarative sentence. Its truth value is false.
c. The sentence “How are you?” is a question and not a declarative sentence. Thus, it is not a proposition.
d. is not a proposition. This is known as an open statement and can be a proposition if we give values for
x. It is true for and it is false for any other values of . For any given value of , it is true or false but not
both.
a. Open the door. c. In the year 2020, the president of the United States will be a woman. b.
7055 is a large number. d. .
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
George Boole (the one who published The Mathematical Analysis of Logic in 1848) used symbols such as p, q,
r, and s to represent propositions and the symbols and to represent connectives. See Table
c. The number 10 is not a prime number / The number 10 is a composite number.
d. The fire engine is red.
Solution
a. b. c. d.
Your turn 3
Use p, q, r, and s as defined in Example 3 to write the following compound propositions in symbolic
form.
Solution
a. The game will be shown on CBS and the game will be played in Atlanta.
b. The game will be shown on ESPN and the Dodgers are favored to win.
c. The Dodgers are favored to win if and only if the game will not be played in Atlanta.
14
PANGASINAN STATE UNIVERSITY
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
T T T
T F T
F T T
F F F
Example 5 Determine the truth value of the following propositions.
a.
b. 5 is a whole number and 5 is an even number.
c. 2 is a prime number and 2 is an even number.
Solution
a. means 7 > 5 or 7 5. Because 7 > 5 is true, the statement 7 5 is a true statement. b.
This is false because 5 is not an even number.
c. This is true because each simple statement is true.
T T T
T F F
F T T
F F T
Solution
a. Because the consequent is true, this is a true statement.
15
PANGASINAN STATE UNIVERSITY
Study Guide in Mathematics in the Modern World FM-AA -CIA-15 Rev. 0 10-July-2020
b. In row 2 of the above truth table, we see that is true when p is true, q is true,
and r is false.
Your turn 9 b. Use the truth table that you constructed in part a to determine the truth value of given that
p is false, q is true, and r is false.
LEARNING POINTS
The term logic refers to the science that studies the principle of correct reasoning. Logic requires the act of
reasoning to form thoughts and opinions, as well as classification and judgments. The foundation of logical
argument is its proposition or statement. The proposition is either accurate (true) or not accurate (false) but
not both true and false. The argument is then built on premises. The premises are the propositions used to
build the argument.
The truth value of a simple statement is either true (T) or false (F).
The truth value of a compound statement depends on the truth values of its simple statements and its
connectives. A truth table is a table that shows the truth value of a compound statement for all possible
truth values of its simple statements.
LEARNING ACTIVITY 3
In exercises 1 to 5. Determine which of the following sentences are propositions and indicate their truth values.
1. Legazpi is the capital of Albay.
2.
3.
for every pair of real numbers and
5. if
5. Answer this question.
a. Because 140 students like volleyball and 85 like both sports, there must be students who like only volleyball.
b. Because 120 students like basketball and 85 like both sports, there must be students who like only
basketball.
c. The Venn diagram shows that the number of students who like only volleyball plus the number who like only
basketball plus the number who like both sports is 55 + 35 + 85=175. Thus of the 200 students surveyed, only do
not like either of the sports.
3. a. c.
b. d.
4. a. True. A conjunction of two statements is true provided that both statements are true.
b. True. A disjunction of two statements is true provided that at least one statement is true.
c. False. If both statements of a disjunction are false, then the disjunction is false.
5 . a. Because the antecedent is true and the consequent is false, the statement is a false statement.
b. Because the antecedent is false, the statement is a true statement.
c. Because the consequent is true, the statement is a true statement.
6. a. Let . Then the first inequality of the biconditional is false, and the second inequality
of the biconditional is true. Thus the given biconditional statement is false.
b. Both inequalities of the biconditional are true for , and both inequalities are false for .
Because both inequalities have the same truth value for any real number x, the given biconditional is true.
7. a.
T T F F F T T row 1 row 2
T F F T T F T row 3 row 4
F T T F F T T
F F T T F T T
21
PANGASINAN STATE UNIVERSITY