Prove That A − B, A∩B and B − A Are Pairwise Disjoint Prove That A − B, A∩B and B − A Are Pairwise Disjoint Question: If \( A \) and \( B \) are sets, then prove that \[ A-B,\quad A\cap B,\quad B-A \] are pairwise disjoint. Solution To prove that the sets […]

Prove That A∩B = Φ ⟺ A ⊆ B’ Prove That A∩B = Φ ⟺ A ⊆ B’ Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap B=\phi \iff A\subseteq B’ \] Solution First, prove that \( A\cap B=\phi \Rightarrow A\subseteq B’ \) Assume: \[ A\cap B=\phi \]

Find Sets A, B and C Such That A∩B∩C = Φ Find Sets A, B and C Such That A∩B∩C = Φ Question: Find sets \( A \), \( B \) and \( C \) such that \[ A\cap B,\quad A\cap C,\quad B\cap C \] are non-empty sets and \[ A\cap B\cap C=\phi \] Solution

Prove That A ∩ (A ∪ B) = A Prove That A ∩ (A ∪ B) = A Question: For any two sets \( A \) and \( B \), prove that: \[ A\cap(A\cup B)=A \] Solution Let \( x\in A\cap(A\cup B) \). Then, \[ x\in A \] and \[ x\in A\cup B \] Since

Prove That A ∪ (A ∩ B) = A Prove That A ∪ (A ∩ B) = A Question: For any two sets \( A \) and \( B \), prove that: \[ A\cup(A\cap B)=A \] Solution Let \( x\in A\cup(A\cap B) \). Then either \[ x\in A \] or \[ x\in A\cap B \]

Prove That A ⊂ B ⇒ C − B ⊂ C − A Prove That A ⊂ B ⇒ C − B ⊂ C − A Question: For three sets \( A \), \( B \) and \( C \), show that: \[ A\subset B \Rightarrow C-B\subset C-A \] Solution Given: \[ A\subset B \]

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 \(