Prove That A = (A ∩ B) ∪ (A − B) Prove That A = (A ∩ B) ∪ (A − B) Question: Show that for any sets \( A \) and \( B \), \[ A=(A\cap B)\cup(A-B) \] Solution \[ (A\cap B)\cup(A-B) \] \[ =(A\cap B)\cup(A\cap B’) \] \[ =A\cap(B\cup B’) \] \[ =A\cap

Is P(A) ∪ P(B) = P(A∪B)? Is P(A) ∪ P(B) = P(A∪B)? Question: Is it true that for any sets \( A \) and \( B \), \[ P(A)\cup P(B)=P(A\cup B) \] Justify your answer. Solution The statement is false. Take \[ A=\{1\}, \quad B=\{2\} \] \[ P(A)=\{\phi,\{1\}\} \] \[ P(B)=\{\phi,\{2\}\} \] \[ P(A)\cup P(B)=\{\phi,\{1\},\{2\}\}

Prove That B’ ⊂ A’ ⇒ A ⊂ B Prove That B’ ⊂ A’ ⇒ A ⊂ B Question: For any two sets \( A \) and \( B \), prove that: \[ B’\subset A’ \Rightarrow A\subset B \] Solution \[ B’\subset A’ \] Taking complement on both sides, \[ (A’)’\subset(B’)’ \] \[ A\subset B

Prove That A’∪B = U ⇒ A ⊂ B Prove That A’∪B = U ⇒ A ⊂ B Question: For any two sets \( A \) and \( B \), prove that: \[ A’\cup B=U \Rightarrow A\subset B \] Solution \[ A’\cup B=U \] Taking complement on both sides, \[ (A’\cup B)’=U’ \] \[ A”\cap

Prove That (A∪B) ∩ (A∪B’) = A Prove That (A∪B) ∩ (A∪B’) = A Question: Using properties of sets, show that for any two sets \( A \) and \( B \), \[ (A\cup B)\cap(A\cup B’)=A \] Solution Consider the left-hand side: \[ (A\cup B)\cap(A\cup B’) \] Using the distributive law, \[ =(A\cup(B\cap B’)) \]

Prove That A − B, A∩B and B − A Are Pairwise Disjoint Prove That A − B, A∩B and B − A Are Pairwise Disjoint Question: If \( A \) and \( B \) are sets, then prove that \[ A-B,\quad A\cap B,\quad B-A \] are pairwise disjoint. Solution To prove that the sets

Prove That A∩B = Φ ⟺ A ⊆ B’ Prove That A∩B = Φ ⟺ A ⊆ B’ Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap B=\phi \iff A\subseteq B’ \] Solution First, prove that \( A\cap B=\phi \Rightarrow A\subseteq B’ \) Assume: \[ A\cap B=\phi \]

Find Sets A, B and C Such That A∩B∩C = Φ Find Sets A, B and C Such That A∩B∩C = Φ Question: Find sets \( A \), \( B \) and \( C \) such that \[ A\cap B,\quad A\cap C,\quad B\cap C \] are non-empty sets and \[ A\cap B\cap C=\phi \] Solution

Prove That A ∩ (A ∪ B) = A Prove That A ∩ (A ∪ B) = A Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap(A\cup B)=A \] Solution Let \( x\in A\cap(A\cup B) \). Then, \[ x\in A \] and \[ x\in A\cup B \] Since

Prove That A ∪ (A ∩ B) = A Prove That A ∪ (A ∩ B) = A Question: For any two sets \( A \) and \( B \), prove that: \[ A\cup(A\cap B)=A \] Solution Let \( x\in A\cup(A\cap B) \). Then either \[ x\in A \] or \[ x\in A\cap B \]