Question

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

Short answer

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

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:

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

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

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

Thus, write as follows:

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

Thus, \(LHS = RHS\)

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