08 Direct Proof and Counterexample + Proof by Contradiction

find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false.

To disprove a statement of the form "∀x ∈ D, if P(x) then Q(x)," by counterexample ....

An integer n is composite if, and only if, n > 1 and n = rs for some integers r and s and r ≠ 1 and s ≠ 1 n is composite ⇔ ∃ positive integers r and s such that n = rs and 1 < r < n and 1 < s < n.

An integer n is composite if, and only if, _____

An integer n is even if, and only if, n equals twice some integer. n=2k

An integer n is even if, and only if, ________ n is even ⇔ ∃ an integer k such that n = 2k.

An integer n is odd if, and only if, n equals twice some integer plus 1. n=2k+1

An integer n is odd if, and only if, _________ ∃ an integer k such that n = 2k + 1.

An integer n is prime if, and only if, n > 1 and for all positive integers r and s, if n = rs, then r=1 or s=1 n is prime ⇔ ∀ positive integers r and s, if n = rs then either r = 1 and s = n or r = n and s = 1.

An integer n is prime if, and only if, ________

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).

If a and b are integers, is 10a + 8b + 1 odd?

Yes, 6a2b = 2(3a2b), and since a and b are integers, so is 3a2b (being a product of integers).

If a and b are integers, is 6a2b even?

No. A prime number is required to be greater than 1. An integer n is prime if, and only if, n > 1 and for all positive integers r and s, if n = rs, then either r or s equals n

Is 1 prime?

Method of Direct Proof 1. Express the statement to be proved in the form "∀x ∈ D, if P(x) then Q(x)." (This step is often done mentally.) 2. Start the proof by supposing x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true. (This step is often abbreviated "Suppose x ∈ D and P(x).") 3. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference.

Method of Direct Proof (3 steps)

To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property.

Method of Generalizing from the Generic Particular

Method of Proof by Contradiction 1. Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true. 2. Show that this supposition leads logically to a contradiction. 3. Conclude that the statement to be proved is true.

Method of Proof by Contradiction 3 steps

Let n = 10. Then 10 = 5 + 5 = 3 + 7 and 3, 5, and 7 are all prime numbers.

Prove the following: ∃ an even integer n that can be written in two ways as a sum of two prime numbers.

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

Suppose that r and s are integers. Prove the following: ∃ an integer k such that 22r + 18s = 2k.

pic3

The sum of any rational number and any irrational number is irrational. Prove by contradiction.

pic1

The sum of any two even integers is even. Prove

pic2

There is no greatest integer. Prove by contradiction.

2, 3, 5, 7, 11, 13

Write the first six prime numbers