Set theory

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Problem: Prove the identity using the mutual inclusion method and Euler-Venn diagrams. \[ \begin{array}{l} A \cup(B \cap C)=(A \cup B) \cap(A \cup C), \\ (A \cup B) \cdot C=(A \cdot C) \cup(B \cdot C) . \end{array} \]

6.8.1 Set theory

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Problem: Write down the set in gray.

6.8.2 Set theory

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Problem: Prove that: \[ \left(A_{1} \cap \ldots \cap A_{n}\right)-\left(B_{1} \cap \ldots \cap B_{n}\right) \subseteq\left(A_{1} \doteq B_{1}\right) \cup \ldots \cup\left(A_{n} \doteq B_{n}\right) \text {. } \]

6.8.3 Set theory

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Problem: Prove that: \[ U^{2} \backslash(A \times B)=[(U \backslash A) \times U] \cup[U \times(U \backslash B)] \text {. } \]

6.8.4 Set theory

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Problem: Prove that there exists such a maximal (by inclusion) set \( A \subseteq \mathbb{R} \) that \[ \forall a, a^{\prime} \in A, a-a^{\prime} \notin Q \text { when } a \neq a^{\prime} \text {. } \]

6.8.5 Set theory

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Problem: Prove set relations: a) \( A \cap B \subset A \cup B \); b) \( A \backslash(A \backslash B)=A \cap B \); c) \( A \backslash C \subset(A \backslash B) \cup(B \backslash C) \); d) \( A \cap(B \cup C)=(A \cap B) \cup(A \cap C) \);

6.8.6 Set theory

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Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. a) \( A((A \cap B) \backslash C)=(A \backslash B) \cup(A \cap C) \) b) \( U^{2} \backslash(C \times D)=(U \times(U \backslash D)) \cup((U \backslash C) \times U) \).

6.8.7 Set theory

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Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. a) \( A \cup(B \backslash C)=(A \cup B) \backslash(C \backslash A) \), b) \( A \times(B \cap C)=(A \times B) \backslash(A \times(B \backslash C)) \).

6.8.8 Set theory

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Problem: Prove the identity: \( \overline{B \backslash A}=\bar{B} \cup A \).

6.8.9 Set theory

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Problem: Prove equality using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. \[ (A \cup B) \backslash(A \cap C)=(A \cap \bar{C}) \cup(\bar{A} \cap B) \text {. } \]

6.8.10 Set theory

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Problem: Prove equalities using the properties of set operations and definitions of operations. \[ C \subseteq D \Rightarrow A \times C \subseteq B \times D \text {. } \]

6.8.11 Set theory

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Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using EulerVenn diagrams. a) \( A \backslash B=A \Delta(A \cap B) \), b) \( (A \cup C) \times B=(C \times B) \cup((A \cap C) \times B) \cup(A \times B) \).

6.8.12 Set theory

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Problem: Prove the equalities. Illustrate using Euler-Venn diagrams. a) \( (A \cap B) \backslash(A \cap C)=(A \cap B) \backslash C \), b) \( (A \cup B) \times C=(A \times C) \cup(B \times C) \).

6.8.13 Set theory

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