Question

Prove the second absorption law from Table 1 by showing that if \(A\) and \(B\) are sets, then \(A \cap \left( {A \cup B} \right) = A\).

Short answer

Thus, it is proved that \(A \cap \left( {A \cup B} \right) = A\).

Step-by-step solution

Given in the question.

Let the sets be \(A = \left\{ {1,2,3} \right\}\) and \(B = \left\{ {4,5,6} \right\}\).

Thus, write as follows:

\(\begin{aligned}{c}A \cup B = \left\{ {1,2,3} \right\} \cup \left\{ {4,5,6} \right\}\\ = \left\{ {1,2,3,4,5,6} \right\}\end{aligned}\)

Showing that \(A \cap \left( {A \cup B} \right) = A\).

Now, above obtained set is \(A \cup B = \left\{ {1,2,3,4,5,6} \right\}\).

Thus, write as follows:

\(\begin{aligned}{c}LHS = A \cap \left( {A \cup B} \right)\\ = \left\{ {1,2,3} \right\} \cap \left\{ {1,2,3,4,5,6} \right\}\\ = \left\{ {1,2,3} \right\}\\ = A\end{aligned}\)

Thus, \(LHS = RHS\)

Hence, it is proved that \(A \cap \left( {A \cup B} \right) = A\).