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    1.4 Predicates and Quantifiers: Predicate Logic

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    1.4 Predicates and Quantifiers: Predicate Logic

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    Công ty Kiếm thẻ cào chúng tôi chia sẻ đến bạn cách kiếm thẻ cào trên điện thoại Android và IOS giúp bạn dễ dàng kiếm thêm thu nhập mỗi ngày mà không mất bất kỳ chi phí nào từ nhiều hình thức khác nhau: làm khảo sát online, đọc báo hay 24h kiếm tiền. Address: Công Ty Kiếm Thẻ Cào Điện Thoại - Xóm Trung Tiến Diễn Hoàng - Diễn Châu - Nghệ An. Phone: (+84) 37 854 4082 Contact Us: Website: https://kiemthecaofree.com/ Facebook: https://www.facebook.com/kiemthecaocom

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    Công ty Kiếm thẻ cào chúng tôi chia sẻ đến bạn cách kiếm thẻ cào trên điện thoại Android và IOS giúp bạn dễ dàng kiếm thêm thu nhập mỗi ngày mà không mất bất kỳ chi phí nào từ nhiều hình thức khác nhau: làm khảo sát online, đọc báo hay 24h kiếm tiền. Address: Công Ty Kiếm Thẻ Cào Điện Thoại - Xóm Trung Tiến Diễn Hoàng - Diễn Châu - Nghệ An. Phone: (+84) 37 854 4082 Contact Us: Website: https://kiemthecaofree.com/ Facebook: https://www.facebook.com/kiemthecaocom

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    1.4 Predicates and Quantifiers: Predicate Logic

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    [INT 1358]: Discrete Mathematics I

    1.4 Predicates and Quantifiers


    Predicate Logic
    Predicate logic have the following features to express propositions:

    • Variables: x, y, z, etc. (the subject of a sentence), can be substituted with an element from a
    domain.

    • Predicates: P, M , etc. (the predicate of a sentence)

    • Domain: the collection of values that a variable can take.

    Propositions must be definitive (not vague or undefined). So, a Propositional Function is not a
    Proposition until all variables are defined (or “bound”).

    Example: Let Q(x, y) denote the statement “x + y > 2x”

    • Q(x, y) has two unbound variables (x and y), and is not a proposition.

    • Q(1, y) = 1 + y > 2 [Not a proposition] one bound variable (x = 1) and one unbound
    variable (y).

    • Q(1, 2) = 1 + 2 > 2 [Proposition] two bound variables (x = 1 and y = 2)

    Quantifiers
    Quantifiers provide a notation that allows us to quantify (count) how many objects in the universe
    of discourse satisfy the given predicate.

    • Universal Quantifier ∀ - For all elements

    • Existential Quantifier ∃ - There exists an element

    De Morgan’s Law for Quantifiers


    • ¬∀xP (x) ≡ ∃x¬P (x)

    • ¬∃xP (x) ≡ ∀x¬P (x)

    1.4 pg. 53 # 5
    Let P (x) be the statement “x spends more than five hours every weekday in class,” where the
    domain for x consists of all students. Express each of these quantifications in English.

    a) ∃xP (x)
    There exists a student who spends more than five hours every weekday in class.

    1
    [INT 1358]: Discrete Mathematics I

    b) ∀xP (x)
    Every student spends more than five hours every weekday in class.

    c) ∃x¬P (x)
    There exists a student who does not spend more than five hours every weekday in class.

    d) ∀x¬P (x)
    No student spends more than five hours every weekday in class.

    1.4 pg. 53 # 11
    Let P (x) be the statement “x = x2 .” If the domain consists of all the integers, what are these truth
    values?

    a) P (0)
    True

    c P (2)
    False

    e ∃xP (x)
    True

    f ∀xP (x)
    False

    1.4 pg. 54 # 25
    Translate each of these statements into logical expressions using predicates, quantifiers, and logical
    connectives. The domain of x is all people.

    c All your friends are perfect.


    Let F (x) be “x is your friend” and P (x) be “x is perfect.”
    ∀x(F (x) → P (x))

    d At least one of your friends is perfect.


    Let F (x) be “x is your friend” and P (x) be “x is perfect.”
    ∃x(F (x) ∧ P (x))

    2
    [INT 1358]: Discrete Mathematics I

    1.4 pg. 55 # 33
    Express each of these statements using quantifiers. Then form the negation of the statement, so
    that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not
    simply use the phrase “It is not the case that.”)

    b No rabbit knows calculus.


    C(x) =“x knows calculus”.
    Domain for x is all rabbits.
    ∀x¬C(x)
    ∃xC(x)
    There exists a rabbit that knows calculus.

    c Every bird can fly.


    F (x) =“x can fly”.
    Domain for x is all birds.
    ∀xF (x)
    ∃x¬F (x)
    There exists a bird who cannot fly.

    1.4 pg. 55 # 35
    Find a counterexample, if possible, to these universally quantified statements, where the domain
    for all variables consists of all integers.

    a ∀x(x2 ≥ x)
    No counter example. See Example 13 in section 1.4.

    b ∀x(x > 0 ∨ x < 0)


    0. Since 0 is not less than or greater than 0, 0 is a counter example.

    c ∀x(x = 1)
    2. Since 2 is not 1, 2 is a counter example.

    1.4 pg. 56 # 59
    Let P (x), Q(x), and R(x) be the statements “x is a professor,” “x is ignorant,” and “x is vain,”
    respectively. Express each of these statements using quantifiers; logical connectives; and P (x),
    Q(x), and R(x), where the domain consists of all people.

    a No professors are ignorant.


    ∀x(P (x) → ¬Q(x))

    b All ignorant people are vain.


    ∀x(Q(x) → R(x))

    3
    [INT 1358]: Discrete Mathematics I

    c No professors are vain.


    ∀x(P (x) → ¬R(x))

    d Does (c) follow from (a) and (b)?


    No because we can have vain professors.

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