Prove That A − (A − B) = A ∩ B Prove That A − (A − B) = A ∩ B Question: For any two sets \( A \) and \( B \), prove that: \[ A-(A-B)=A\cap B \] Solution \[ A-(A-B) \] \[ =A\cap(A-B)’ \] \[ =A\cap(A\cap B’)’ \] \[ =A\cap(A’\cup B) \] […]

Prove That A ∩ (A’ ∪ B) = A ∩ B Prove That A ∩ (A’ ∪ B) = A ∩ B Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap(A’\cup B)=A\cap B \] Solution \[ A\cap(A’\cup B) \] \[ =(A\cap A’)\cup(A\cap B) \] \[ =\phi\cup(A\cap B) \]

Prove That A’ − B’ = B − A Prove That A’ − B’ = B − A Question: For any two sets \( A \) and \( B \), prove that: \[ A’-B’=B-A \] Solution \[ A’-B’ \] \[ =A’\cap(B’)’ \] \[ =A’\cap B \] \[ =B\cap A’ \] \[ =B-A \] Hence proved.

Find the Value of n Using Sets and Counting Principle Find the Value of n Using Sets and Counting Principle Question: Each set \( X_r \) contains 5 elements and each set \( Y_r \) contains 2 elements and \[ \bigcup_{r=1}^{20}X_r=S=\bigcup_{r=1}^{n}Y_r \] If each element of \( S \) belongs to exactly 10 of the

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

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, \[ =(A\cup(B\cap B’)) \]