Math 340, Homework 2.
The homework is intended to give you significantly more practice with set theoretical notation, especially the set builder
notation.
The setCraft tool understands two kinds of set builder notation: First, the type described by the text, where we write
{x∈A; [statement about x]}. The technical name for this construction is comprehension. Perhaps that’s not a very good
name. The second flavor of set builder notation is the form {f(x); x∈A}. This is subtly different, and incorporates the
idea of doing something to each of the elements of A. The construction is called replacement. When we refer to set builder
notation, we mean either kind.
Go to setCraft, and accomplish these things. As always, check the right hand window to make sure you’re doing what
you think you’re doing! In most of these problems, your goal is to construct a set. It’s never sufficient to construct it directly
by listing its elements, because the point is to understand the set builder mechanism.
(1) Define a set Nwhich will be all of the integers from 1 to 100.
(2) From N, use set builder notation to select just those integers greater than 50.
(3) From N, use set builder notation to express the numbers that are perfect squares. (Hint: The perfect squares are
numbers whose square roots are in N. Square roots can be written in terms of exponentiation.)
(4) From N, use set builder notation to get just the odd numbers. (There are lots of ways to do it, but all of them
require some thought!)
(5) From your previous answers, construct the set of odd squares.
(6) From your previous answers, construct the set of even squares.
(7) Build the set {10,20, . . . , 100}in some way more clever than listing all of its elements.
(8) Build the set {2,7,12,17,...,97}in some way more clever than listing all of its elements.
(9) Build the set of all of the numbers from N, except the unlucky numbers 13 and 4.
(10) First let S={1..8}. Find a way to build, from S, the set
{{1},{1,2},{1,2,3},{1,2,3,4},{1,2,3,4,5},{1,2,3,4,5,6},{1,2,3,4,5,6,7},{1,2,3,4,5,6,7,8}}
without, of course, simply listing its elements, as I have. Use set builder notation.
(11) Test setCraft’s opinion about whether the empty set is a subset of {1,3,5}.