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Chapter 4 Unions and Intersections
Exercises Practice Problems
1. Making Sets From Other Sets.
Let \(A= \{ x \mid 1 < x < 5 \}\text{,}\) \(B = \{ x \mid 5 \leq x \leq 7 \}\) and \(C= \{ x \mid 2 < x < 8 \}\text{,}\) where \(x\) represents a real number. Write the following sets in "set builder" notation.
\(A \cup C \text{.}\)
\(A \cap B\text{.}\)
\((A \cup B ) \cap C\text{.}\)
\(A \cap \overline{C}\text{.}\)
2. Union and Intersection.
The concepts of union and intersection naturally correspond to the concepts of "or" and "and." Let's make this more concrete. Let \(P(x)\) and \(Q(x)\) be statements, and define \(A = \{ x \mid P(x) \mbox{ is true} \}\) and \(B = \{ x \mid Q(x) \mbox{ is true} \}\text{.}\) Find a simple way to write each of \(A \cup B\) and \(A \cap B\) using set builder notation. Next, how would you write \(A \backslash B\text{?}\)
3. DeMorgan's Law.
Consider the following DeMorgan Law: \[ \overline{(A \cap B)} = \overline{A} \cup \overline{B}. \]
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Prove this law using a Venn diagram
Prove this law using a carefully argued double inclusion proof.
4. Distributive Laws.
Prove the following distributive laws in two ways. First, using a Venn diagram, and then using a double inclusion proof.
\(\displaystyle A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
\(\displaystyle A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)