How do you prove that every integer is even or odd?

How do you prove that every integer is even or odd?

An integer n is said to be even if it can be expressed in the form n = 2k for some integer k, and odd if it can be expressed as n = 2l + 1 for some integer l. Theorem 85. Every integer is either even or odd, but not both.

How do you prove that N 2 n is even?

Prove: If n is an even integer, then n2 is even. – If n is even, then n = 2k for some integer k. – n2 = (2k)2 = 4k2 – Therefore, n = 2(2k2), which is even.

How do you write a proof by contradiction?

We follow these steps when using proof by contradiction:

1. Assume your statement to be false.
2. Proceed as you would with a direct proof.
3. Come across a contradiction.
4. State that because of the contradiction, it can’t be the case that the statement is false, so it must be true.

How do you prove something is always even?

If is an integer (a whole number), then the expression represents an even number, because even numbers are the multiples of 2. The expressions 2 n − 1 and 2 n + 1 can represent odd numbers, as an odd number is one less, or one more than an even number.

How do you prove all statements?

Following the general rule for universal statements, we write a proof as follows:

1. Let be any fixed number in .
2. There are two cases: does not hold, or. holds.
3. In the case where. does not hold, the implication trivially holds.
4. In the case where holds, we will now prove . Typically, some algebra here to show that .

How do you prove that N 2 N is not an odd number?

How can we proof it? Theorem: If n is an odd integer, then n2 is an odd integer. Proof: Since n is an odd integer, there exists an integer k such that n=2k+1. Therefore, n2 = (2k+1)2 = 4k2+4k+1 = 2(2k2+2k)+1.

How do you disprove a statement in discrete mathematics?

To disprove the original statement is to prove its negation, but a single example will not prove this “for all” statement. The point made in the last example illustrates the difference between “proof by example” — which is usually invalid — and giving a counterexample.

Do the proof by contradiction to prove that the sum of two even integers is even?

That means is not an integer. “Since the sum of two even numbers 2a and 2b must always be an integer that’s divisible by 2, this contradicts the supposition that the sum of two even numbers is not always even. Hence, our original proposition is true: the sum of two even numbers is always even.”

How do you prove a statement is false?

A counterexample disproves a statement by giving a situation where the statement is false; in proof by contradiction, you prove a statement by assuming its negation and obtaining a contradiction.

How do you prove that a negative integer is even?

Statement: “The negative of any even integer is even.” Now let r = -k. Then r is an integer, r = −k= (−1) k. Hence, −n = 2r for some integer r.