Discrete Mathematics - 4.1

Pataasin ang iyong marka sa homework at exams ngayon gamit ang Quizwiz!

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


Kaugnay na mga set ng pag-aaral

Ch 7 & 8 Exam Practice Questions

View Set

NurseLogic Testing and Remediation Beginner

View Set

Physical Science PHS111 Test 03 Study Guide

View Set