Section 6.2 Properties of Sets
Now that we have defined operations on sets such as union and intersection, we can look at various properties of these operations. In the last section we saw that intersection involves an “and” statement, while union involves an “or.” As we work with these properties we will be able to see connections between properties of sets and the logical connectives we saw in Chapter 2.
One of our goals in this section is to learn how to prove properties such as the following subset relations.
Proving Subset.
Based in their definitions from Section 6.1, we can translate other sets into logical statements, as well. These translations are how we will prove properites of sets using elements.
Translating Set Statements to Logical Statements.
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if and only if or -
if and only if and -
if and only if and -
if and only if -
if and only if and
Activity 6.2.1.
Prove Be sure to write what you want to assume and what you want to show.
Activity 6.2.2.
Prove Be sure to write what you want to assume and what you want to show.
Activity 6.2.3.
Proving Set Equality.
Example 6.2.2. Proving Two Sets are Equal.
Prove
Proof.
Therefore,
Therefore,
Therefore, (again by the second subset relation).
The proofs in Example 6.2.1 and Example 6.2.2 are called element arguments. This type of proof uses elements and the definitions of sets and subsets. You start with being an element of the set of interest, use what you know about the set to then get as an element of another set. This technique generalizes to sets in all different mathematical contexts.
The next theorem shows that the empty set is a subset of every set.
Theorem 6.2.3.
Proof.
There are many times in mathematics that we need to prove that a set is empty. We might need do do this if we need sets to be disjoint, or if we need to prove that there are no elements with a particular property. The common method for proving a set is empty is to use contradiction. Note, usually if we want to prove we show both subsets ( ), but we just showed always. So we just need to show We do this by contradiction.
Proving a Set is Empty.
Example 6.2.4. Proving a Set is Empty.
Prove where is the universal set.
Activity 6.2.4.
Prove Be sure to write what you want to assume and what you want to show.
Activity 6.2.5.
Prove if then Note, you need to show So how do you show a subset? What set should you assume is in?
Summary of Set Identities.
Let be the universal set, and and subsets of the universal set.
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Commutative Laws.
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Associative Laws.
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Distributive Laws.
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Identity Laws.
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Complement Laws.
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Double Complement Law.
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Idempotent Laws.
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Universal Bound Laws.
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DeMorgan’s Laws.
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Absorption Laws.
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Complements.
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Set Difference Law.
Reading Questions Check Your Understanding
Let be sets.
1.
True.
False.
True or false:
2.
True.
False.
True or false:
3.
True.
False.
True or false:
4.
True.
False.
True or false:
5.
True.
False.
True or false:
6.
True.
False.
True or false:
7.
True.
False.
8.
True.
False.
True or false:
9.
True.
False.
True or false:
Exercises Exercises
1.
Fill in the blanks.
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To say that an element is in
means that it is in ___ and in ___. -
To say that an element is in
means that it is in ___ or in ___. -
To say that an element is in
means that it is in ___ and not in ___.
2.
Illustrate the distributive laws by drawing the Venn diagrams for the given sets.
3.
Illustrate DeMorgan’s laws by drawing the Venn diagrams for the given sets.
4.
Use an element argument to prove for all sets and Be sure to write what you want to assume and what you want to show.
5.
Use an element argument to prove for all sets and Be sure to write what you want to assume and what you want to show.
6.
Use an element argument to prove for all sets and if then Be sure to write what you want to assume and what you want to show.
7.
Use an element argument to prove for all sets and if then Be sure to write what you want to assume and what you want to show.
8.
Use an element argument to prove for all sets and if then Be sure to write what you want to assume and what you want to show.
9.
Hint.
10.
11.
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