10. Let A, B, and C be sets. Use the identities to show that \( (A \cup B) \cap (B \cup C) \cap (A \cup C) = \bar{A} \cap \bar{B} \cap \bar{C} \).
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A ∪ B ∩ B ∪ C ∩ A ∪ C = (A ∪ B) ∩ (B ∪ C) ∩ (A ∪ C) Show more…
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Let $A, B,$ and $C$ be sets. Use the identities in Table 1 to show that $\overline{(A \cup B)} \cap \overline{(B \cup C)} \cap \overline{(A \cup C)}=\overline{A} \cap \overline{B} \cap \overline{C}$
Basic Structures: Sets, Functions, Sequences, Sums,and Matrices
Set Operations
(9) Show that the equality (A ∩ B) ∪ C = A ∩ (B ∪ C) is not always valid. (10) Prove the following: (a) A ⊆ B iff A ∪ B = B. (b) A ⊆ B iff A ∩ B = A. (c) If A ⊆ B ∪ C and A ∩ B = ∅, then A ⊆ C. (d) If A ⊆ C and B ⊆ C, then A ∪ B ⊆ C. (e) If A ∪ B ⊆ C ∪ D, C ⊆ A, and A ∩ B = ∅, then B ⊆ D.
Madhur L.
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