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If [tex]A = \{a, b, c\}[/tex], [tex]B = \{b, c, d, e\}[/tex], and [tex]C = \{c, e, f, g\}[/tex], verify the following distributive laws.

a) [tex]A \cap (B \cup C) = (A \cap B) \cup (A \cap C)[/tex]

b) [tex]A \cup (B \cap C) = (A \cup B) \cap (A \cup C)[/tex]

Sagot :

GinnyAnswer
Sure, let's verify the distributive laws using the given sets [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].

Given:
[tex]\[ A = \{a, b, c\} \][/tex]
[tex]\[ B = \{b, c, d, e\} \][/tex]
[tex]\[ C = \{c, e, f, g\} \][/tex]

### Part (a): [tex]\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)[/tex]

1. Calculate [tex]\( B \cup C \)[/tex]:
[tex]\[ B \cup C = \{b, c, d, e\} \cup \{c, e, f, g\} = \{b, c, d, e, f, g\} \][/tex]

2. Calculate [tex]\( A \cap (B \cup C) \)[/tex]:
[tex]\[ A \cap (B \cup C) = \{a, b, c\} \cap \{b, c, d, e, f, g\} = \{b, c\} \][/tex]

3. Calculate [tex]\( A \cap B \)[/tex]:
[tex]\[ A \cap B = \{a, b, c\} \cap \{b, c, d, e\} = \{b, c\} \][/tex]

4. Calculate [tex]\( A \cap C \)[/tex]:
[tex]\[ A \cap C = \{a, b, c\} \cap \{c, e, f, g\} = \{c\} \][/tex]

5. Calculate [tex]\( (A \cap B) \cup (A \cap C) \)[/tex]:
[tex]\[ (A \cap B) \cup (A \cap C) = \{b, c\} \cup \{c\} = \{b, c\} \][/tex]

Now, we compare:
[tex]\[ A \cap (B \cup C) = \{b, c\} \][/tex]
[tex]\[ (A \cap B) \cup (A \cap C) = \{b, c\} \][/tex]

Since both sides are equal, the distributive law is verified:
[tex]\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \][/tex]

### Part (b): [tex]\( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)[/tex]

1. Calculate [tex]\( B \cap C \)[/tex]:
[tex]\[ B \cap C = \{b, c, d, e\} \cap \{c, e, f, g\} = \{c, e\} \][/tex]

2. Calculate [tex]\( A \cup (B \cap C) \)[/tex]:
[tex]\[ A \cup (B \cap C) = \{a, b, c\} \cup \{c, e\} = \{a, b, c, e\} \][/tex]

3. Calculate [tex]\( A \cup B \)[/tex]:
[tex]\[ A \cup B = \{a, b, c\} \cup \{b, c, d, e\} = \{a, b, c, d, e\} \][/tex]

4. Calculate [tex]\( A \cup C \)[/tex]:
[tex]\[ A \cup C = \{a, b, c\} \cup \{c, e, f, g\} = \{a, b, c, e, f, g\} \][/tex]

5. Calculate [tex]\( (A \cup B) \cap (A \cup C) \)[/tex]:
[tex]\[ (A \cup B) \cap (A \cup C) = \{a, b, c, d, e\} \cap \{a, b, c, e, f, g\} = \{a, b, c, e\} \][/tex]

Now, we compare:
[tex]\[ A \cup (B \cap C) = \{a, b, c, e\} \][/tex]
[tex]\[ (A \cup B) \cap (A \cup C) = \{a, b, c, e\} \][/tex]

Since both sides are equal, the distributive law is verified:
[tex]\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \][/tex]

Both distributive laws are indeed verified as true.
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