Educational

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 \[ A\cap(A\cup B)’ \] \[ =A\cap A’\cap B’ \] \[ =(A\cap A’)\cap B’ \] \[ =\phi\cap B’ \]

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