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Blast into Math! ets of nummers: mathematical plaagrounds

3.6 Examples and hints

• Hint for # 2: You can do # 2 using only the definitions.
• Hint for # 3: You can use # 2 to prove # 3. There are other correct but different proofs too!

• Hint for # 4: All a, b ∈ N are at least as big as 1,

a ≥ 1, b ≥ 1.

So, what you’re trying to do is find some n ∈ N so that when you multiply it with a , you get
something BIGGER than b. Remember: N is closed under addition and multiplication. So,
starting with a and b , you can add or multiply them together (or with other natural numbers),
and you’ll end up with a natural number. Play around with a and b and see if you can use
them to build an n which solves the problem. And, remember that both a ≥ 1 and b ≥ 1.

• Hint for # 5: Try a proof by induction. What’s the base case? That is n =1, so the set has
just one element. It looks like this:
S = {a}.

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