310 Review Questions and Answers
Question 11-7
Is it possible to multiply a quantity by a positive real number and have the product be smaller
than the original quantity? If so, give an example.Answer 11-7
Yes, this is possible. If the original quantity is negative, then when we multiply it by a positive
real number, it gets more negative. That means it gets smaller. Remember that numbers get
smaller, not larger, as they grow more negative!Question 11-8
Is it possible to divide a positive real number by another positive real number, and have the
quotient turn out strictly larger than the original? If so, give an example.Answer 11-8
Yes, this can happen. Suppose we start with a positive real number, and then divide it by
some positive number smaller than 1. That will give us a quotient larger than the original. For
example, if we divide 3 by 1/4, we get 12, which is larger than 3.Question 11-9
Suppose we have a “strictly larger than” inequality. What can we do to both sides, and be sure
that the result will be another valid “strictly larger than” statement?Answer 11-9
We can do any of the following things:- Add the same quantity to both sides.
- Subtract the same quantity from both sides.
- Multiply both sides by the same positive quantity.
- Divide both sides by the same positive quantity.
Question 11-10
Imagine a hypothetical relation called clobber (symbolized by ©) and three variables a,b, and c.
How do we define the reflexive, symmetric, and transitive properties for the clobber relation?
If clobber has all three of these properties, what sort of relation is it?Answer 11-10
Clobber is reflexive if and only if, for all possible values of a,a©aClobber is symmetric if and only if, for all possible values of a and b,(a©b)⇒ (b©a)Clobber is transitive if and only if, for all possible values of a,b, and c,[(a©b) and (b©c)]⇒ (a©c)If clobber has all three of these properties, then clobber is an equivalence relation.