Question

Prove the second associative law from Table 1 by showing that if A, B, and C are sets, then A ∩ (B ∩ C) = (A ∩ B) ∩ C.

Short answer

Thus, it is true \(\left( {A \cap B} \right) \cap C = A \cap \left( {B \cap C} \right)\).

Step-by-step solution

Given

Given in the question A, B, and C are set. now we proof here,

\(\left( {A \cap B} \right) \cap C = A \cap \left( {B \cap C} \right)\).

solution

Here we solve our question,

\(\left( {A \cap B} \right) \cap C = A \cap \left( {B \cap C} \right)\)

Solve the LHS part ,

\(x \in \left( {A \cap B} \right) \cap C\)

\( \Rightarrow x \in A\)and \(x \in B\)and\(x \in C\).

= (\(x \in A\) and \(x \in B\)) and\(x \in C\).

\( \Rightarrow x \in \left( {A \cap B} \right)\)and\(x \in C\).

\( \Rightarrow x \in \left( {A \cap B} \right) \cap C\)

\( \Rightarrow A \cap \left( {B \cap C} \right) \subseteq \left( {A \cap B} \right) \cap C\)________(i)

Now we solve part RHS

Let \(y \in \left( {A \cap B} \right) \cap C\).

\( \Rightarrow y \in A\)and \(y \in B\)and\(y \in C\)

\( \Rightarrow y \in A\)and (\(y \in B\) and \(y \in C\))

\( \Rightarrow y \in A\)and \(y \in \left( {B \cap A} \right)\).

\( \Rightarrow y \in A \cap \left( {B \cap A} \right)\)

\( \Rightarrow \left( {A \cap B} \right) \cap C \subseteq A \cap \left( {B \cap A} \right)\)__________________(ii)

Frome equation (i) and (ii)

\(\left( {A \cap B} \right) \cap C = A \cap \left( {B \cap A} \right)\)

Hence , LHS = RHS

\(\left( {A \cap B} \right) \cap C = A \cap \left( {B \cap A} \right)\).