Prove the following statements. (a) If n is odd, then n^2 is odd. (Hint: If n is odd, then there

Prove the following statements. (a) If n is odd, then n^2 is...

Key Concepts

Definition of Odd Numbers

An integer is defined as odd if it can be expressed in the form 2k + 1 for some integer k. This algebraic representation encapsulates the intrinsic properties of odd numbers, making it easier to apply in proofs and manipulations, such as demonstrating that operations like squaring preserve oddness.

Definition of Even Numbers

An integer is defined as even if it can be written as 2k for some integer k. This concept is essential in many proofs, particularly when proving implications by considering the negation of an odd number, which directly translates to being even.

Direct Proof

A direct proof establishes a result by starting from the given assumption and using a series of logical steps, often involving algebraic manipulations and known definitions, to arrive at the conclusion. In the case of proving that an odd number squared remains odd, the direct manipulation of the expression obtained from the definition of odd numbers is used.

Proof by Contrapositive

Proof by contrapositive is a method where instead of proving an implication directly, one proves that the negation of the conclusion implies the negation of the hypothesis. This approach is equivalent to proving the original implication and is particularly useful when the contrapositive form is easier to handle, such as proving that if n is even then n^2 must also be even, which in turn proves that if n^2 is not even (i.e., odd) then n is not even (i.e., odd).

Transcript

00:01 Okay, so here in part a, we want to prove that if n is odd, then n squared is odd.

00:08 Well, if n is odd, then we know that there exists some integer k such that n is equal to 2k plus 1, right? because the number is even if it's equal to 2k for some integer k.

00:23 And if it's odd, well, then it's one more than an even number.

00:26 Therefore, it's going to be equal to 2k plus 1.

00:29 Okay.

00:31 So if we just plug this in for n squared, what do we get? so if n is odd, then n is equal to 2k plus 1.

00:39 Therefore, n squared is going to be equal to 2k plus 1 squared, which is going to be equal to 2k plus 1 times 2k plus 1 gives us a 4k squared plus 4k plus 1.

00:57 But, ooh, we can actually factor this as the 4k squared plus 4k just factors as 2 times k squared plus 2k, right? that's two, that's two times 2k squared.

01:13 That's going to be a 4k squared plus 4k plus 1.

01:18 But then we can just go ahead and replace because 2k squared plus k is just some integer.

01:24 Just call that some integer.

01:26 Maybe some integer, i don't know, j.

01:29 So therefore that n squared is going to be equal to, well, just two times some integer, call that it to be j.

01:38 So 2j plus 1, which shows here that n squared is just equal to two times an integer plus 1, right? j, right, call jk if you want.

01:47 It doesn't matter, right? you just equal to n squared is equal to two times some integer plus 1.

01:51 This shows that n squared is odd.

01:57 So therefore, if n is odd, we then conclude that n squared must also be odd as well.

02:05 For part b, we want to prove the statement.

02:10 Well, if n squared is odd, then n is going to be odd.

02:20 Okay, well, we can prove this statement by finding...

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