In Class Activity
MAT 300, Spring 2004
February 5, 2004
Set Equivalence
Prove that the set described by “the intersection of E and F” is the same set as that
described by “the set of all natural numbers that are multiples of 4.” I.e. the two sets are
equivalent, so is a multiple of 4}.
xxFE |{ ℵ∈=∩
State clearly a definition of set equivalence that matches your proof.
Describe a procedure that can be used to show that two given sets are equivalent
(assuming that they are equivalent).
Consider Other Sets
Now let us return to the three sets from last week: A, B, C that have non-empty
intersections.
1. Try to label as many of the sets as possible with more than one label.
2. Pick a space where you have two descriptions for the same space. Try to prove that
those two ways of describing the space are equivalent.
3. Would your two expressions from 2 always be equivalent for any sets A, B, C? Does
it matter whether or not A is a subset of B or if B and C are disjoint?
Here are some informal notes from our book about two other useful sets:
• The difference A\B is the set of all elements in A that are not in B.
• The symmetric difference A∆B is the set of elements that are either in A or B, but not
both.
Find two ways of writing each of these sets in terms of compliments, intersections and
unions.