Find Number of Elements in Q Find Number of Elements in Q Question: If \( P \) and \( Q \) are two sets such that \[ n(P)=40,\quad n(P\cup Q)=60,\quad n(P\cap Q)=10 \] how many elements does \( Q \) have? Solution \[ n(P\cup Q)=n(P)+n(Q)-n(P\cap Q) \] \[ 60=40+n(Q)-10 \] \[ 60=30+n(Q) \] \[ n(Q)=60-30 […]

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

Find n(A∩B) If n(A∪B)=50, n(A)=28 and n(B)=32 Find n(A∩B) Question: If \( A \) and \( B \) are two sets such that \[ n(A\cup B)=50,\quad n(A)=28,\quad n(B)=32 \] find \( n(A\cap B) \). Solution \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] \[ 50=28+32-n(A\cap B) \] \[ 50=60-n(A\cap B) \] \[ n(A\cap B)=60-50 \] \[ n(A\cap B)=10

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

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: \[ A-(A\cap B)=A-B \] Solution \[ A-(A\cap B) \] \[ =A\cap(A\cap B)’ \] \[ =A\cap(A’\cup B’) \] \[ =(A\cap

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

Prove That If A ⊂ B Then C − B ⊂ C − A Prove That If A ⊂ B Then C − B ⊂ C − A Question: If \( A \), \( B \), \( C \) are three sets such that \( A\subset B \), then prove that: \[ C-B\subset C-A \]

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

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