Predicates and Quantified Statements x is an even number x is a student in Discrete Math x+y=5 x is the son of z These are not statements. Why? When the variable(s) takes specific values, they become propositions. The sets were the variable takes values is called the domain D. Each sentence describe properties hold by the variable(s) involve in it. These sentences are called Predicates or Open Statements.

Notation for Predicates P(x): x is an even number Possible domains: N, Z, R Q(y): y is a student in Discrete Math Possible domains: TAMUCC, College of Science R(x, y): x + y =5 Possible domains: N, Z, Q, R S(x, z): x is the son of z Possible domains: People in CC, TX, USA, the world

True Set P(x): x is an even number If D is the domain, represents all the elements in the domain that make P(x) true. If D={1,2,3,4} The true set is { 2,4} If D=N, the true set is {0,2,4,6,8,…}

Let Q(n) be the predicate “n is a factor of 8 Let Q(n) be the predicate “n is a factor of 8.” Find the truth set of Q(n) if the domain of n is the set Z+ of all positive integers {1, 2, 4, 8} b. the domain of n is the set Z of all integers. {1, 2, 4, 8,−1,−2,−4,−8}

Quantifiers Consider P(x): x is an even number with domain all integers How to make a predicate a proposition? By substituting the variable by a specific value: P(3), P(4), etc. All integers satisfy P(x) Some integers satisfy P(x)

P(x): x is an even number To make a predicate into a proposition two components are needed: the quantifier and the domain (where the variable takes values) P(x): x is an even number All integers satisfy P(x) Quantifiers Domain Some integers satisfy P(x)

P(x): x is an even number Equivalent Expressions for “All” P(x): x is an even number All integers satisfy P(x) for every x, P(x) for arbitrary x, P(x) for any x, P(x) for each x, P(x) given any x, P(x) Let D = {1, 2, 3, 4, 5}, and consider the statement Show that this statement is true. Let D = {-1, 0.25, 0.5, 1}, and consider the statement Show that this statement is false. (counterexample)

P(x): x is an even number Equivalent Expressions for “Some” P(x): x is an even number Some integers satisfy P(x) there is an a, P(a), we can find a, P(a), there is at least one a, P(a) for some a, P(a), for at least one a, P(a) Let D = {-1, 0.25, 0.5, 1}, and consider the statement Show that this statement is false.

From Formal to Informal Rewrite the following formal statements in a variety of equivalent but more informal ways. Do not use the symbol ∀ or ∃. a. b. c.

Solutions a. All real numbers have nonnegative squares. Or: Every real number has a nonnegative square. Or: Any real number has a nonnegative square. Or: The square of each real number is nonnegative. b. All real numbers have squares that are not equal to −1. Or: No real numbers have squares equal to −1. (The words none are or no . . . are are equivalent to the words all are not.)

c. There is a positive integer whose square is equal to itself c. There is a positive integer whose square is equal to itself. Or: We can find at least one positive integer equal to its own square. Or: Some positive integer equals its own square. Or: Some positive integers equal their own squares.

Rewrite the following statement informally, without quantifiers or variables. ∀x ∈ R, if x > 2 then x2 > 4. Solution: If a real number is greater than 2 then its square is greater than 4. Or: Whenever a real number is greater than 2, its square is greater than 4. Or: The square of any real number greater than 2 is greater than 4. Or: The squares of all real numbers greater than 2 are greater than 4.

Equivalent Forms of Universal and Existential Statements Observe that the two statements “∀ real numbers x, if x is an integer then x is rational” and “∀ integers x, x is rational” mean the same thing. Both have informal translations “All integers are rational.” In fact, a statement of the form can always be rewritten in the form by narrowing U to be the domain D consisting of all values of the variable x that make P(x) true.

Exercise Rewrite the statement "All squares are rectangles" in the two forms “∀x, if ______ then ______” and “∀ ______x, _______” Solution: ∀x, if x is a square then x is a rectangle. ∀ squares x, x is a rectangle.

On the other hand, a statement of the form “∃x such that P(x) and Q(x)” can be rewritten as “∃x εD such that Q(x),” where D is the set of all x for which P(x) is true.

A statement of the form “∃x such that P(x) and Q(x)” can be rewritten as “∃x εD such that Q(x),” where D is the set of all x for which P(x) is true.

A prime number is an integer greater than 1 whose only positive integer factors are itself and 1. Consider the statement “There is an integer that is both prime and even.” Let P (n) be “n is prime” and E(n) be “n is even.” Use the notation P (n) and E (n) to rewrite this statement in the following two forms: a. ∃n such that ______ ∧ ______ . b. ∃ ______ n such that ______.

NEGATION OF “ALL” “SOME” All the students in Discrete Math are math majors Statement in symbolic form Let D be the Discrete Math students, M(x): x is a math major Negation of the original statement Some discrete math students are not math majors Negation in symbolic form

Some students in Discrete Math wear glasses Statement in symbolic form D Discrete Math students, W(x): x wears glasses Negation of original statement Any Discrete Math student does not wear glasses Negation in symbolic form

SUMMARY OF NEGATIONS FOR "ALL " "SOME"

Contrapositive, Converse, Inverse of a Universal Implication Consider a statement of the form Contrapositive Converse Inverse

Example Write a formal and an informal contrapositive, converse, and inverse for the statement: If a real number is greater than 2, then its square is greater than 4.

Contrapositive: ∀x ∈ R, if x2 ≤ 4 then x ≤ 2 Contrapositive: ∀x ∈ R, if x2 ≤ 4 then x ≤ 2. If the square of a real number is less than or equal to 4, then the number is less than or equal to 2. Converse: ∀x ∈ R, if x2 > 4 then x > 2. If the square of a real number is greater than 4, then the number is greater than 2. Inverse: ∀x ∈ R, if x ≤ 2 then x2 ≤ 4. If a real number is less than or equal to 2, then the square of the number is less than or equal to 4.

Necessary, Sufficient, Only if For all x, P(x) is a sufficient condition for Q(x) For any x, Q(x) is a necessary condition for P(x) For any x, P(x) only if Q(x)

Example Rewrite the following statements as quantified conditional statements. Do not use the word necessary or sufficient. a. Squareness is a sufficient condition for rectangularity. b. Being at least 35 years old is a necessary condition for being President of the United States.

Solution a. A formal version of the statement is ∀x, if x is a square, then x is a rectangle. In informal language: If a figure is a square, then it is a rectangle. b. A formal version ∀ people x, if x is younger than 35, then x cannot be President of the United States. Or, by the equivalence between a statement and its contrapositive: ∀ people x, if x is President of the United States, then x is at least 35 years old. Informal version Any president of the United States is at least 35 years old