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Three sets \( A, B, \) and \( C \) are represented as:

[tex]\[ A = \{x \in \mathbb{N} : x \leq 7\} \][/tex]
[tex]\[ B = \{2, 3, 5, 7\} \][/tex]
[tex]\[ C = \{1, 3, 5, 7, 9\} \][/tex]

Show that:

(a) \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)

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

Sagot :

GinnyAnswer
Sure, let's go through each part of the problem step-by-step.

### Part (a) Show that \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)

1. Identify Sets:
[tex]\[ \begin{aligned} &A = \{x \in \mathbb{N} : x \leq 7\} = \{1, 2, 3, 4, 5, 6, 7\} \\ &B = \{2, 3, 5, 7\} \\ &C = \{1, 3, 5, 7, 9\} \end{aligned} \][/tex]

2. Calculate \( B \cap C \):
[tex]\[ B \cap C = \{2, 3, 5, 7\} \cap \{1, 3, 5, 7, 9\} = \{3, 5, 7\} \][/tex]

3. Calculate \( A \cup (B \cap C) \):
[tex]\[ A \cup (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\} \cup \{3, 5, 7\} = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]

4. Calculate \( A \cup B \):
[tex]\[ A \cup B = \{1, 2, 3, 4, 5, 6, 7\} \cup \{2, 3, 5, 7\} = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]

5. Calculate \( A \cup C \):
[tex]\[ A \cup C = \{1, 2, 3, 4, 5, 6, 7\} \cup \{1, 3, 5, 7, 9\} = \{1, 2, 3, 4, 5, 6, 7, 9\} \][/tex]

6. Calculate \( (A \cup B) \cap (A \cup C) \):
[tex]\[ (A \cup B) \cap (A \cup C) = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 2, 3, 4, 5, 6, 7, 9\} = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]

7. Comparison:
[tex]\[ A \cup (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]
[tex]\[ (A \cup B) \cap (A \cup C) = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]

Thus, we have shown:
[tex]\[ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \][/tex]

### Part (b) Show that \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)

1. Calculate \( B \cup C \):
[tex]\[ B \cup C = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7, 9\} = \{1, 2, 3, 5, 7, 9\} \][/tex]

2. Calculate \( A \cap (B \cup C) \):
[tex]\[ A \cap (B \cup C) = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 2, 3, 5, 7, 9\} = \{1, 2, 3, 5, 7\} \][/tex]

3. Calculate \( A \cap B \):
[tex]\[ A \cap B = \{1, 2, 3, 4, 5, 6, 7\} \cap \{2, 3, 5, 7\} = \{2, 3, 5, 7\} \][/tex]

4. Calculate \( A \cap C \):
[tex]\[ A \cap C = \{1, 2, 3, 4, 5, 6, 7\} \cap \{1, 3, 5, 7, 9\} = \{1, 3, 5, 7\} \][/tex]

5. Calculate \( (A \cap B) \cup (A \cap C) \):
[tex]\[ (A \cap B) \cup (A \cap C) = \{2, 3, 5, 7\} \cup \{1, 3, 5, 7\} = \{1, 2, 3, 5, 7\} \][/tex]

6. Comparison:
[tex]\[ A \cap (B \cup C) = \{1, 2, 3, 5, 7\} \][/tex]
[tex]\[ (A \cap B) \cup (A \cap C) = \{1, 2, 3, 5, 7\} \][/tex]

Thus, we have shown:
[tex]\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \][/tex]

This completes the solution.
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