Download DISCRETE MATH: LECTURE 4 1. Chapter 3.1 Predicates and

DISCRETE MATH: LECTURE 4
DR. DANIEL FREEMAN
1. Chapter 3.1 Predicates and Quantified Statements I
• A predicate is a sentence that contains a finite number of variables and becomes
a statement when specific values are substituted for the variables. The domain of
a predicate variable is the set of all values that may be substituted in place of the
variable.
For Example: Consider the sentences ”Daniel is a Fightin’ Texas Aggie,” and ”Daniel
is a graduate of Texas A&M University.”
• If P (x) is a predicate and x has domain D, the truth set of P (x) is the set of all
elements of D that make P (x) true when they are substituted for x. The truth set
of P (x) is denoted: {x ∈ D|P (x)}.
For Example: Let P (x) be the predicate ”x2 > x” with domain R. Write P (2), P ( 21 ),
and P (− 12 ), and indicate which of these statements are true and which are false.
1.1. The Universal Quantifier: ∀.
• Quantifiers are words that refer to quantities such as ”some” or ”all” and tell for
how many elements a given predicate is true.
• The symbol ∀ denotes ”for all” and is called the universal quantifier.
• Let Q(x) be a predicate and D the domain of x. A universal statement is a
statement of the form ”∀x ∈ D, Q(x).” It is defined to be true if, and only if, Q(x)
is true for every x in D. It is defined to be false if, and only if, Q(x) is false for at
least one x in D. A value for x for which Q(x) is false is called a counterexample
to the universal statement.
For Example: We can express the sentence ”All human beings are mortal,” as ∀
human beings x, x is mortal.
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Your Example: Write down a universal statement. Define the domain and its truth
set. Then, write down a false universal statement, give the domain, and give a
counterexample.
1.2. The Existential Quantifier: ∃.
• The symbol ∃ denotes ”there exists” and is called the existential quantifier.
• Let Q(x) be a predicate and D the domain of x. An existential statement is a
statement of the form ”∃x ∈ D such that Q(x).” It is defined to be true if, and
only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false
for all x in D.
For Example: ∃m ∈ Z+ such that m2 = m.
For Example: Let E = {5, 6, 7, 8}. ∃m ∈ E such that m2 = m.
1.3. Universal Conditional Statements.
• A universal conditional statement has the form: ∀x, if P (x) then Q(x).
For Example: Rewrite each of the following statements in the form: ∀
then
.
(1) If a real number is an integer, then it is a rational number.
(2) All bytes have eight bits.
(3) No fire trucks are green.
, if
DISCRETE MATH: LECTURE 4
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1.4. Equivalent Forms of Universal and Existential Statements.
∀x ∈ U, if P (x) then Q(x) is equivalent to
∀x ∈ D, Q(x)
∃x such that P (x) and Q(x) is equivalent to ∃x ∈ D such that Q(x).
2. Predicates and Quantified Statements II: The Quantifiers Strike Back
• The negation of a statement of the form ∀x in D, Q(x) is logically equivalent to a
statement of the form ∃x in D such that ∼ Q(x). Symbolically,
∼ (∀x ∈ D, Q(x)) ≡ ∃x ∈ D such that ∼ Q(x).
• The negation of a universal statement (”all are”) is logically equivalent to an existential statement (”some are not” or ”there is at least one that is not”).
• When we speak of logical equivalence for quantified statements, we mean
that the statements always have identical truth values no matter what predicates
are substituted for the predicate symbols and not matter what sets are used for the
domains of the predicate variables.
• The negation of a statement of the form ∃x in D such that Q(x) is logically equivalent to a statement of the form ∀x in D, ∼ Q(x). Symbolically,
∼ (∃x ∈ D such that Q(x)) ≡ ∀x ∈ D, ∼ Q(x).
• The negation of an existential statement (”some are” is logically equivalent to a
universal statement (”none are” or ”all are not”).
2.1. In Class Work. Determine whether the proposed negation is correct. If it is not,
write a correct negation.
Statement
Proposed Negation
1 The sum of any two irrational numbers The sum of any two irrational numbers is rational.
numbers is irrational
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For all integers n,
For all integers n, if n2 is even then n is not even.
if n2 is even then n is even.
2.2. Negations of Universal Conditional Statements.
• The negation of a universal conditional statement is
∼ (∀x, if P (x) then Q(x)) ≡ ∃x such that P (x) and ∼ Q(x).
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DR. DANIEL FREEMAN
2.3. In Class Work. Write a negation for each statement:
(1) ∀ real numbers x, if x2 ≥ 1 then x > 0.
(2) ∀x ∈ R, if x(x + 1) > 0 then x > 0 or x < −1.
(3) ∀ integers a, b, and c, if a − b is even and b − c is even, then a − c is even.
(4) If the square of an integer is odd, then the integer is odd.
2.4. The Relation Among ∀, ∃, ∧, and ∨.
• In a sense, universal statements are generalizations of and statements, and existential statements are generalizations of or statements.
• If Q(x) is a predicate and the domain D of x is the set {x1 , x2 , . . . , x3 }, then the
statements
∀x ∈ D, Q(x)
and
Q(x1 ) ∧ Q(x2 ) ∧ · · · ∧ Q(xn )
are logically equivalent.
• Similarly, if Q(x) is a predicate and D = {x1 , x2 , . . . , xn }, then the statements
∃x ∈ D such that Q(x)
and
Q(x1 ) ∨ Q(x2 ) ∨ · · · ∨ Q(xn )
are logically equivalent.
• This is why the negation of statements with the quantifiers ”for all” and ”there
exists” are analogous to the De Morgan’s Laws.
2.5. Vacuous Truth of Universal Statements.
• In general, a statement of the form
∀x in D, if P (x) then Q(x)
is called vacuously true or true by default if, and only if, P (x) is false for every
x in D
Your Example: Write down a universal statement that is vacuously true.
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2.6. Variants of Universal Conditional Statements.
Consider a statement of the form: ∀x ∈ D, if P (x) then Q(x).
Its contrapositive is the statement ∀x ∈ D, if ∼ Q(x) then ∼ P (x)
Its converse is the statement
∀x ∈ D, if Q(x) then P (x).
Its inverse is the statement
∀x ∈ D, if ∼ P (x) then ∼ Q(x)
2.7. In Class Work. Write the contrapositive, converse, and inverse of the following
statement:
If a real number is greater than 2, then its square is greater than 4.
Show that a universal conditional statement is logically equivalent to its contrapositive.
Show that a universal conditional statement is NOT logically equivalent to its converse.
Show that a universal conditional statement is NOT logically equivalent to its inverse.
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DR. DANIEL FREEMAN
2.8. Necessary and Sufficient Conditions, Only If.
• ”∀x, r(x) is a sufficient condition for s(x)” means ”∀x, if r(x) then s(x).”
• ”∀x, r(x) is a necessary condition for s(x)” means ”∀x, if ∼ r(x) then ∼ s(x)”
or, equivalently, ”∀x, if s(x) then r(x).”
• ”∀x, r(x) only if s(x)” means ”∀x, if s(x) then r(x)”, or equivalently, ”∀x, if r(x)
then s(x).”
2.9. In Class Work. Rewrite the following statements as quantified conditional statements. Do not use the word necessary or sufficient.
(1) Squareness is a sufficient condition for rectangularity.
(2) Being at least 35 years old is a necessary condition for being President of the United
States.
Rewrite the following as a universal conditional statement:
A product of two numbers is 0 only if one of the numbers is 0.