集合 $A = \{1, 2\}$、$B = \{3\}$、$C = \{4, 5\}$ に対して、次の集合を求め、その濃度を求めよ。 (1) $B \times A$ (2) $A \times B \times C$ (3) $(A \times B) \times C$ (4) $A \times (B \times C)$

1. 問題の内容

集合 $A = \{1, 2\}$、$B = \{3\}$、$C = \{4, 5\}$ に対して、次の集合を求め、その濃度を求めよ。

(1) $B \times A$

(2) $A \times B \times C$

(3) $(A \times B) \times C$

(4) $A \times (B \times C)$

2. 解き方の手順

(1) $B \times A$

$B \times A = \{(x, y) \mid x \in B, y \in A\}$

$B \times A = \{(3, 1), (3, 2)\}$

濃度は $|B \times A| = |B| \times |A| = 1 \times 2 = 2$

(2) $A \times B \times C$

$A \times B \times C = \{(x, y, z) \mid x \in A, y \in B, z \in C\}$

$A \times B \times C = \{(1, 3, 4), (1, 3, 5), (2, 3, 4), (2, 3, 5)\}$

濃度は $|A \times B \times C| = |A| \times |B| \times |C| = 2 \times 1 \times 2 = 4$

(3) $(A \times B) \times C$

$A \times B = \{(1, 3), (2, 3)\}$

$(A \times B) \times C = \{((1, 3), 4), ((1, 3), 5), ((2, 3), 4), ((2, 3), 5)\}$

濃度は $|(A \times B) \times C| = |A \times B| \times |C| = |A| \times |B| \times |C| = 2 \times 1 \times 2 = 4$

(4) $A \times (B \times C)$

$B \times C = \{(3, 4), (3, 5)\}$

$A \times (B \times C) = \{(1, (3, 4)), (1, (3, 5)), (2, (3, 4)), (2, (3, 5))\}$

濃度は $|A \times (B \times C)| = |A| \times |B \times C| = |A| \times |B| \times |C| = 2 \times 1 \times 2 = 4$

3. 最終的な答え

(1) $B \times A = \{(3, 1), (3, 2)\}$、濃度は2

(2) $A \times B \times C = \{(1, 3, 4), (1, 3, 5), (2, 3, 4), (2, 3, 5)\}$、濃度は4

(3) $(A \times B) \times C = \{((1, 3), 4), ((1, 3), 5), ((2, 3), 4), ((2, 3), 5)\}$、濃度は4

(4) $A \times (B \times C) = \{(1, (3, 4)), (1, (3, 5)), (2, (3, 4)), (2, (3, 5))\}$、濃度は4