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Exercises 2.2.3 Exercises
1.
Give the definition for these terms. Be sure to set up any notation that is required.
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2.
State precisely De Morgan’s laws for sets. Be sure to set up any notation that is required.
3.
Let \(C = \set{r, s, t, l, n, e}\text{,}\) and let \(W =\set{f,o,r,t,u,n,e}\text{.}\) Compute each of these sets.
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\(\displaystyle C \cap W\)
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\(\displaystyle C \cup W\)
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\(\displaystyle C\lxor W\)
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4.
Let
\(A\text{,}\) \(B\text{,}\) \(C\) be sets. Shade the portion of the Venn diagram corresponding to each of these sets.
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\(\displaystyle A\cup(B\cap C)\)
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\(\displaystyle A\cap(B\cap C)\)
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\(\displaystyle A\cap(B\cup C)\)
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\(\displaystyle (\overline{A} \cap \overline{B}) \cup \overline{C}\)
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\(\displaystyle (A - B) - C\)
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\(\displaystyle (A - B) \cup (A - C) \cup (B - C)\)
5.
Let \(A = \set{1, 3, 5, 7, 9}\text{,}\) and let \(B = \set{1, 2, 3, 4, 5}\text{.}\) Compute the following sets.
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\(\displaystyle A \cap B\)
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\(\displaystyle A \cup B\)
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\(\displaystyle B\lxor A\)
6.
Let
\(A\) and
\(B\) be sets. Prove that if
\(A \subseteq B\text{,}\) then
\(A \cup B = B\text{.}\)
7.
8.
9.
10.
Prove De Morgan’s laws for sets.
11.
Let
\(A\) and
\(B\) be sets. Prove
\(A\lxor B = (A - B) \cup (B - A)\text{.}\)
12.
Let \(A\) and \(B\) be sets.
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Prove if
\(A \subseteq B\) then
\(\overline{B} \subseteq \overline{A}\text{.}\)
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Prove if
\(\overline{B} \subseteq \overline{A}\) then
\(A \subseteq B\text{.}\)
