Discrete Mathematics - 4.1

What are the first 10 prime numbers?

2,3,4,7,11,13,17,19,23,29

Proving Existential Statements

According to the definition given in Section 3.1, a statement in the form: ∃x ∈ D such that Q(x) is true if, and only if, Q(x) is true for at least one x in D.

An integer n is _______ if, and only if, n equals twice some integer.

Even _________________________________________ Symbolically, if n is an integer, then: n is even ⇔ ∃ an integer k such that n = 2k.

An integer n is ____________ if, and only if, n equals twice some integer plus 1.

Odd _________________________________________ Symbolically, if n is an integer, then: n is odd ⇔ ∃ an integer k such that n = 2k + 1.

Disproving Universal Statements by Counterexample

To disprove a statement of the form: "∀x ∈ D, if P(x) then Q(x)," find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called a counterexample.

Proving Universal Statements

We are trying to prove "∀x ∈ D, if P(x) then Q(x)" or "∀x ∈ D, P(x)" _________________________________________ 1. method of exhaustion: works if the domain D is a finite set: then we can prove the property P(x) is true for every element in the set one by one _________________________________________ 2. When the domain is an infinite set; we use "method of generalizing from the generic particular" _________________________________________ • We assume X is a particular but arbitrarily chosen element of the set D and we show that x satisfies the property. _________________________________________ • If we apply this proof to a universal conditional statement then the method becomes "direct proof".

disproving a existential statement

We have "∃x ∈ D such that P(x)". If we want to disprove it; we will prove it's negation is true: "∀x ∈ D, ~p(x)."

Constructive Proofs of Existence: a. Prove the following: ∃ an even integer n that can be written in two ways as a sum of two prime numbers. b. Suppose that r and s are integers. Prove the following: ∃ an integer k such that 22r + 18s = 2k.

a. Let n = 10. Then 10 = 5 + 5 = 3 + 7 and 3 , 5, and 7 are all prime numbers. _________________________________________ b. Let k = 11r + 9s. Then k is an integer because it is a sum of products of integers; and by substitution, 2k = 2(11r + 9s), which equals 22r + 18s by the distributive law of algebra.

Prime and Composite Numbers: a. Is 1 prime? b. Is every integer greater than 1 either prime or composite? c. Write the first six prime numbers. d. Write the first six composite numbers.

a. No. A prime number is required to be greater than 1. _________________________________________ b. Yes. Let n be any integer that is greater than 1. Consider all pairs of positive integers r and s such that n = rs. There exist at least two such pairs, namely r = n and s = 1 and r = 1 and s = n. Moreover, since n = rs, all such pairs satisfy the inequalities 1 ≤ r ≤ n and 1 ≤ s ≤ n.If n is prime, then the two displayed pairs are the only ways to write n as rs. Otherwise, there exists a pair of positive integers r and s such that n = rs and neither r nor s equals either 1 or n. Therefore, in this case 1 < r < n and 1 < s < n, and hence n is composite. _________________________________________ c. 2,3,5,7,11,13 _________________________________________ d. 4,6,8,9,10,12

Even and Odd Integers: a. Is 0 even? b. Is −301 odd? c. If a and b are integers, is 6a^2b even? d. If a and b are integers , is 10a + 8b + 1 odd?

a. Yes, 0=2·0. b. Yes, −301 = 2(−151) + 1. c. Yes, 6a^2b = 2(3a^2b), and since a and b are integers, so is 3a^2b (being a product of integers). d. Yes, 10a + 8b + 1 = 2(5a+4b) +1, and since a and b are integers, so is 5a + 4b (being a sum of products of integers).

An integer n is prime ___________

if, and only if, n > 1 and ∀ r, s ∈ Z+ ; if n = r*s, then either r = n or s =n

An integer n is composite ________

if, and only if, n > 1 if n is not prime. N is composite <=> ∃ r, s ∈ Z+ such that n= r*s and 1 < r < n and 1 < s < n.

In symbols: n is prime ⇔ ___________

∀ r, s ∈ Z+ ; if n = r*s, then either r = n or s =n

In symbols: n is composite ⇔ ___________

∃ r, s ∈ Z+ such that n= r*s and 1 < r < n and 1 < s < n