Educational

Find How Many Like Both Coffee and Tea Find How Many Like Both Coffee and Tea Question: In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea? Solution \[ n(C\cup T)=n(C)+n(T)-n(C\cap T) \] \[ 70=37+52-n(C\cap […]

Find How Many Teachers Teach Physics Find How Many Teachers Teach Physics Question: In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach physics and mathematics. How many teach physics? Solution \[ n(M\cup P)=n(M)+n(P)-n(M\cap P) \] \[ 20=12+n(P)-4 \] \[ 20=8+n(P) \] \[ n(P)=20-8 \]

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: \[ C-B\subset C-A \]