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.

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 =5Possible domains: N, Z, Q, R

S(x, z): x is the son of zPossible domains: People in CC, TX, USA, the world

Notation for Predicates

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.” Find the truth set of Q(n) if

a. 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}

Consider P(x): x is an even number with domain all integersHow to make a predicate a proposition?1. By substituting the variable by a specific

value: P(3), P(4), etc. 2. All integers satisfy P(x)

3. Some integers satisfy P(x)

Quantifiers

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 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)

Equivalent Expressions for “All”

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.

Equivalent Expressions for “Some”

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.

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 majorsStatement in symbolic formLet 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 majorsNegation in symbolic form

Some students in Discrete Math wear glassesStatement in symbolic formD Discrete Math students, W(x): x wears glasses

Negation of original statementAny Discrete Math student does not wear glassesNegation 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.

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