Beginning Activity Beginning Activity 1: Exploring a Relationship between Two Sets
Let \(A\) and \(B\) be subsets of some universal set \(U\text{.}\)
1.
Draw two general Venn diagrams for the sets \(A\) and \(B\text{.}\) On one, shade the region that represents \(\left( {A \cup B} \right)^c\text{,}\) and on the other, shade the region that represents \(A^c \cap B^c\text{.}\) Explain carefully how you determined these regions.
2.
Based on the Venn diagrams in Exercise 1, what appears to be the relationship between the sets \(\left( {A \cup B} \right)^c\) and \(A^c \cap B^c\text{?}\)
Some of the properties of set operations are closely related to some of the logical operators we studied in Section 2.1. This is due to the fact that set intersection is defined using a conjunction (and), and set union is defined using a disjunction (or). For example, if \(A\) and \(B\) are subsets of some universal set \(U\text{,}\) then an element \(x\) is in \(A \cup B\) if and only if \(x \in A\) or \(x \in B\text{.}\)
3.
Use one of De Morgan's Laws (Theorem 2.12 to explain carefully what it means to say that an element \(x\) is not in \(A \cup B\text{.}\)
4.
What does it mean to say that an element \(x\) is in \(A^c\text{?}\) What does it mean to say that an element \(x\) is in \(B^c\text{?}\)
5.
Explain carefully what it means to say that an element \(x\) is in \(A^c \cap B^c\text{.}\)
6.
Compare your response in Exercise 3 to your response in Exercise 5. Are they equivalent? Explain.
7.
How do you think the sets \(\left( {A \cup B} \right)^c\) and \(A^c \cap B^c\) are related? Is this consistent with the Venn diagrams from Exercise 1?