Logic and Proof Exercises

Example 1

What's the base case when we prove inductively that 3ⁿ + 1 is divisible by 2 for all positive integers?

Hint

Choose the simplest positive integer you can think of, then plug it in for n.

Example 2

What's the base case when we prove inductively that 10ⁿ – 5 is divisible by 5 for all positive integers?

Hint

Use the smallest possible n-value. No need to get fancy.

Example 3

What's the base case when we prove inductively that n – 2 is positive when n > 2?

Hint

We can't use n = 1 this time, so what's the next simplest value we can plug in? Whole numbers are easier than fractions/decimals.

Example 4

Brandon is trying to prove that 3n + 1 is an even number whenever n is a positive integer. You want to mess up his proof, because you think he's wrong. Name one counterexample that shows he can't prove his general statement.

Hint

A counterexample is just any n-value that makes the statement untrue.

Example 5

Are there any counterexamples that disprove the statement, "6n + 2 is always divisible by 4 when n is a positive integer"?

Hint

It works fine when n = 1, but pay close attention to other possible values for n.

Logic and Proof Exercises

Example 1

Example 2

Example 3

Example 4

Example 5

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