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).
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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.

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