Here is quiz 3.
Problem 1 requests a counterexample to the statement: An integer is positive if and only if is positive.
To find a counterexample to a statement of the form “an integer has property if and only if it has property we need an integer such that one of the properties is true of while the other is false. Note that if is positive, i.e., then i.e., is also positive. This means that the only way the statement is going to fail is if we have that is positive yet is not.
We can now easily see that is a counterexample. (And, in fact, it is the only counterexample.)
Problem 2 asks to show that where and
By definition, iff any element of is also an element of Let us then consider an element of By definition of this means that is an integer, and We need to show that i.e., that and that The first requirement is automatic, so we only need to verify that
To see this, we use that By definition, this means that there is an integer, let us call it such that To show that we need (again, by definition) that there is an integer such that Now, since then or We immediately see that we can take This is an integer, and as needed.
Problem 3 asks for the cardinality of the set
Recall that for any set then as shown in lecture. In particular, if , then Now let Then