# MATH135 Lecture Notes - Lecture 7: Open Formula, Mathematical Object, Natural Number

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Published on 29 Sep 2015

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MATH 135 - Lecture 7: Universal and Existential Quantifiers:

Quantifiers:

● Is “5x + 3 ≥ 8” a statement?

○ No, it is sometimes true and sometimes false depending on the variable x

○ If something depends on a variable, it is called an open sentence

○ We can change an open sentence into a statement by adding a quantifier

Ex. 1 “For all x , 5∈ ℤ x + 3 ≥ 8” (otherwise written as ∀x , 5∈ ℤ x + 3 ≥ 8)

→ which is false and is an example of a universal quantifier

Ex. 2 “There exists x , such that 5∈ ℤ x + 3 ≥ 8” (otherwise written as ∃x , ∈ ℤ

5x + 3 ≥ 8) → which is true and is an example of an existential quantifier

Universal Quantifiers:

● ∀x S, P(∈x) where:

○ is the quantifier “for all”∀

○x is the variable

○ S is the domain (universe of discourse)

○ P(x) is the open sentence

○ To prove, we show that P(x) is true for every element in S

● Select method:

○ we select a representative mathematical object x S (but NOT a specific ∈

element)

○ Ex. Prove: For every natural number n, 2n2 + 11n + 15 is composite

Note: composite = has a divisor other than 1 and itself

Proof: Let n ∈ ℕ

2n2 + 11n + 15 = (2n + 5)(n + 3)

Since n , ∈ ℕ n ≥ 1

Therefore 2n + 5 ≥ 7 and n + 3 ≥ 4

Therefore neither factor is 1

Therefore 2n2 + 11n + 15 is composite

● Substitution method:

○ ∀x S may appear in the hypothesis and we may have: If ∈ ∀ x S, P(∈x), then

Q(x)

○ We can use any element from S and P(x) to prove the implication

○ We can substitute any value of x from S into P(x) and use that statement in our

proof

○ Ex. Prove: Let a, b, c . If ∈ ℤ ∀ x , such that ∈ ℤ a | (bx + c), then a | (b + c)

Proof: Assume that ∀x , such that ∈ ℤ a | (bx + c)

1 . Therefore, ∈ ℤ a | (b + c).

## Document Summary

Math 135 - lecture 7: universal and existential quantifiers: No, it is sometimes true and sometimes false depending on the variable x. If something depends on a variable, it is called an open sentence. We can change an open sentence into a statement by adding a quantifier. Which is false and is an example of a universal quantifier. 5x + 3 8) which is true and is an example of an existential quantifier. X + 3 8 (otherwise written as x + 3 8 (otherwise written as. S, p: where: is the quantifier for all . S is the domain (universe of discourse) To prove, we show that p(x) is true for every element in s. We select a representative mathematical object x. Prove: for every natural number n, 2n2 + 11n + 15 is composite. Note: composite = has a divisor other than 1 and itself. Therefore 2n + 5 7 and n + 3 4.