2 Techniques of Proof

2.1 Proof by Contraposition

Objectives:

• Discover proof techniques for basic conditional propositions

• Apply methods of contrapositive and contradiction to several number theory examples

2.1.1 Indirect Proofs

Example 2.1.1.

Construct the contrapositive of the following statements:

• If $n\in \mathbb{Z}$ such that $n^2$ is even, then $n$ is even.

  Blank space for you to write your work.

• If $n,m\in \mathbb{Z}$ such that $nm$ is even, then $n$ is even or $m$ is even.

  Blank space for you to write your work.

Example 2.1.2.

Compare each bulleted statement with its contrapositive statement. Which do you think would be easier to prove? Try writing the first few lines of a proof for each statement to determine.

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Typically it is easier to move from less complex statements to more complex statements when given the choice. In each of the bulleted statements above, the antecedent can be thought of as hiding crucial information. For example, in the first bullet, how would we show that taking a “square root” of $n^2$ retains that $n$ is even for integers $n$? For the second bullet, if $nm$ is even, what steps would we take to “divide by” $m$ and show that $n$ is even?

These mathematical operations seem a bit trickier to manage than the ones used in the contrapositive statements.

Example 2.1.3.

Using the contrapositive statement you wrote in the previous example, prove the following statement: “if $n\in \mathbb{Z}$ such that $n^2$ is even, then $n$ is even.”

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On the previous page we reasoned that the negation of "P OR Q" is "not P AND not Q". We codify this into one of DeMorgan's Laws (I): $$\neg (P\vee Q)⟺\neg P\wedge \neg Q$$

In heuristic language, this means that distributing a negation through a compound statement results in flipping any logical connectors like ∨ or ∧.

Example 2.1.4.

Write a truth table proving DeMorgan's Laws (II):$$\neg (P\wedge Q)⟺\neg P\vee \neg Q$$

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Example 2.1.5.

Using the contrapositive statement you wrote in the previous example, prove the following statement: “if $n,m\in \mathbb{Z}$ such that $nm$ is even, then $n$ is even or $m$ is even.”

Blank space for you to write your work.

2.1.2 Biconditional Proofs

The strongest statements we have seen so far are statements of logical equivalence. Consider the statement:

The integer $n$ is even if and only if $n^2$ is even.

The connecting phrase “if and only if” (sometimes abbreviated iff) can be unpacked the following way: “if either of the two statements on either side of me is true, then the statement on the other side is also true. What’s more, if either of the two statements on either side of me is false, then the statement on the other side is also false.”

This last statement is quite interesting, as it says that we could have very well replaced "even" with "odd" in the statement and have a logically equivalent statement (verify!):

The integer $n$ is odd if and only if $n^2$ is odd.

Example 2.1.6.

Let $P$ be the statement “$n$ is even” and let $Q$ be the statement “$n^2$ is even”. Using a truth table, prove that $P\iff Q$ is logically equivalent to the statement $\neg P\iff \neg Q$.

Blank space for you to write your work.

Example 2.1.7.

Prove the statement: “The integer $n$ is even iff $n^2$ is even.”

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Example 2.1.8.

Disprove the statement: “For an integer $n$, the value $n$ is even iff 4 divides $n$.”

Blank space for you to write your work.