Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

1.

List at least four different positive elements of \(S\) and at least four different negative elements of \(S\text{.}\) Are all of these integers even?

2.

Use the roster method to specify the sets \(S\) and \(T\text{.}\) (See Section 2.3 for a review of the roster method.) Does there appear to be any relationship between these two sets? That is, does it appear that the sets are equal or that one set is a subset of the other set?

3.

Use set builder notation to specify the sets \(S\) and \(T\text{.}\) (See Section 2.3 for a review of the set builder notation.)

4.

Using appropriate definitions, describe what it means to say that an integer \(x\) is a multiple of 6 and what it means to say that an integer \(y\) is even.

5.

In order to prove that \(S\) is a subset of \(T\text{,}\) we need to prove that for each integer \(x\text{,}\) if \(x \in S\text{,}\) then \(x \in T\)

Complete the know-show table in Table 5.11 for the proposition that \(S\) is a subset of \(T\text{.}\)

This table is in the form of a proof method called the choose-an-element method. This method is frequently used when we encounter a universal quantifier in a statement in the backward process. (In this case, this is Step \(Q1\text{.}\)) The key is that we have to prove something about all elements in \(\Z\text{.}\) We can then add something to the forward process by choosing an arbitrary element from the set \(S\text{.}\) (This is done in Step \(P1\text{.}\)) This does not mean that we can choose a specific element of \(S\text{.}\) Rather, we must give the arbitrary element a name and use only the properties it has by being a member of the set \(S\text{.}\) In this case, the element is a multiple of 6.