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演習

1. 集合 $A, B$ に対して、次を示せ。
   (1) $A - (A - B) = A \cap B$
   (2) $(A \cap B) - C = (A - C) \cap (B - C)$

2. 集合 $A, B, C, D$ に対して、次を示せ。
   (1) $A \triangle (A \triangle B) = B$
   (2) $A \triangle B = C \triangle D$ ならば $A \triangle C = B \triangle D$

3. $A = \{2, 4, 6\}, B = \{2, 3, 4, 5, 6\}, E = \{2n \mid n \in \mathbb{N} \}$ とする。
   (1) $A$ は $E$ の部分集合であることを示せ。
   (2) $B$ は $E$ の部分集合でないことを示せ。

4. 集合 $\{a, b, c\}$ の部分集合で、少なくとも要素を1つもつものをすべてあげよ。

5. $\{4n^2 + 6n + 2 \mid n \in \mathbb{Z}\} = \{m^2 + m \mid m \in \mathbb{Z}\}$ を示せ。

Would you like me to solve a specific question or provide explanations for these problems?

Related Questions:

1. How do you prove set identities involving intersections and differences?
2. What is the symmetric difference $A \triangle B$ in set theory, and what are its properties?
3. How do you determine whether a set is a subset of another?
4. What are all subsets of a set with three elements?
5. How do you prove that two sets are equal using set notation?

Tip: To prove equality of two sets $X = Y$, show that $X \subseteq Y$ and $Y \subseteq X$.