Page 64 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

What are the possible subsets? Well, there’s always ∅ , because it’s a subset of every set. What

else? S . Those are the only subsets: how many subsets are there? That’s right, there are 2= 2 .
1
So, the base case is true. Now, we assume the theorem is true for n . This means that the set
which contains n distinct elements, let’s call it S , has 2 subsets. So, now we think about
n
the set which contains n +1 distinct elements; let’s call it S . So, S has all the elements of
∗
∗
S and one new element. How many subsets does S have? First, we can count all the subsets
∗
of S because they’re also subsets of S . How many subsets does S have? It has 2 by the
n
∗
induction assumption. How can we get other subsets of S ? Well, we can take each of these
∗
2 subsets and include the new element. How many subsets does that give us total?
n
• Hint for # 6: First, square several different integers whose last digit is 5. Try to find a
pattern. Then, prove your algorithm by induction or using cases. There could be more than
one right answer…
• Hint for # 7: Follow the same basic steps from the proof of Gauss’s formula.
• Example for # 8: The set of numbers x on the number line such that |x − 3| > 2 is the
set of numbers x whose distance from 3 on the number line is greater than 2. Draw this.
Then, you can write this set as the union of two sets,

{x> 5}∪ {x< 1}.

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