Page 59 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

3. Prove the GLB Proposition.
4. Show that if a, b ∈ N there exists n ∈ N such that na >b.
5. Show that for any n ∈ N , the set S of n distinct elements has exactly 2 subsets
n
including the empty set and S itself.
6. In this exercise, you must find a computational recipe for squaring integers which end in 5,
and prove that it will always give the correct answer. A computational recipe is an example
of an algorithm. In the next chapter, you will learn about a computational recipe known as
the Euclidean algorithm.
7. Summation and Σ. We proved by induction that the sum of all the integers from 1 up to n

where n ∈ N is

n(n +1)
.
2

One way to write the sum of all the integers from 1 up to n is

1+2+ ... + n.

There is a more succinct way to write this.

n

1+2+ ... + n = k.
k=1

The symbol represents summation: it tells us to add. You might be wondering, why is there
k on the right side of the equation but not on the left side? The letter k is used to tell us what
we add. We could also use a different letter like j or i. The top and bottom of Σ tell us which
k ’s we add. The bottom

k=1

means we start with k =1. When do we stop adding? That is explained by the top

n

.

So, we add all the integers from 1 up to n . That is what

n

k
k=1

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