Educational

Show That A ∩ B = A ∩ C Need Not Imply B = C Show That A ∩ B = A ∩ C Need Not Imply B = C Question: For three sets \( A \), \( B \) and \( C \), show that \[ A\cap B=A\cap C \] need not imply \[ […]

Equivalent Statements of A ⊂ B Equivalent Statements of A ⊂ B Question: For any two sets \( A \) and \( B \), show that the following statements are equivalent: \[ (i)\ A\subset B \] \[ (ii)\ A-B=\phi \] \[ (iii)\ A\cup B=B \] \[ (iv)\ A\cap B=A \] Solution We prove the equivalence

Prove That If A ⊂ B Then A ∩ B = A Prove That If A ⊂ B Then A ∩ B = A Question: For any two sets \( A \) and \( B \), prove that: \[ A\subset B \implies A\cap B=A \] Solution Given: \[ A\subset B \] This means every element

Prove That A ∩ B ⊂ A Prove That A ∩ B ⊂ A Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap B \subset A \] Solution Let \( x\in A\cap B \). By the definition of intersection of sets, \( x \) belongs to both \(

Prove That B ⊂ A ∪ B Prove That B ⊂ A ∪ B Question: For any two sets \( A \) and \( B \), prove that: \[ B \subset A\cup B \] Solution Let \( x \in B \). Then \( x \) belongs to set \( B \). By the definition of

Prove That B ⊂ A ∪ B Prove That B ⊂ A ∪ B Question: For any two sets \( A \) and \( B \), prove that: \[ B \subset A\cup B \] Solution Let \( x \in B \). Since \( x \) belongs to set \( B \), it must belong to

Prove That (A∩B)’ = A’ ∪ B’ Prove That (A∩B)’ = A’ ∪ B’ Question: If \[ U=\{2,3,5,7,9\} \] is the universal set, \[ A=\{3,7\}, \quad B=\{2,5,7,9\} \] then prove that: \[ (A\cap B)’=A’\cup B’ \] Solution First find \( A\cap B \): \[ A\cap B=\{7\} \] Now find \( (A\cap B)’ \): \[ (A\cap

Prove That (A∪B)’ = A’ ∩ B’ Prove That (A∪B)’ = A’ ∩ B’ Question: If \[ U=\{2,3,5,7,9\} \] is the universal set, \[ A=\{3,7\}, \quad B=\{2,5,7,9\} \] then prove that: \[ (A\cup B)’=A’\cap B’ \] Solution First find \( A\cup B \): \[ A\cup B=\{2,3,5,7,9\} \] Now find \( (A\cup B)’ \): \[ (A\cup

Verify A∩(B Δ C) = (A∩B) Δ (A∩C) Verify A∩(B Δ C) = (A∩B) Δ (A∩C) Question: Let \[ A=\{1,2,4,5\},\quad B=\{2,3,5,6\},\quad C=\{4,5,6,7\} \] Verify the identity: \[ A\cap(B\Delta C)=(A\cap B)\Delta(A\cap C) \] Solution First find the symmetric difference \( B\Delta C \): \[ B\Delta C=(B-C)\cup(C-B) \] Now, \[ B-C=\{2,3\} \] \[ C-B=\{4,7\} \] Therefore, \[

Verify A − (B∩C) = (A − B)∪(A − C) Verify A − (B∩C) = (A − B)∪(A − C) Question: Let \[ A=\{1,2,4,5\},\quad B=\{2,3,5,6\},\quad C=\{4,5,6,7\} \] Verify the identity: \[ A-(B\cap C)=(A-B)\cup(A-C) \] Solution First find \( B\cap C \): \[ B\cap C=\{5,6\} \] Now find \( A-(B\cap C) \): \[ A-(B\cap C) =