Question

Show that if \(A\) and \(B\) are sets with \(A \subseteq B\), then

(a) \(A \cup B = B\)

(b) \(A \cap B = A\)

Short answer

(a) it is proved that \(A \cup B = B\).

(b) it is proved that \(A \cap B = A\).

Step-by-step solution

(a) Showing that \(A \cup B = B\).

It is given that \(A \subseteq B\). Thus,

\(A \subseteq B = \left\{ {x|x \in A \Rightarrow x \in B} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{a}}\;{\rm{subset}}\)

Solve the left hand side of set \(A \cup B\) as follows:

\(\begin{aligned}LHS &= A \cup B\\ &= \left\{ {x|x \in A \vee x \in B} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{union}}\\ &= \left\{ {x|x \in B \vee x \in B} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{a}}\;{\rm{subset}}\\ &= B\end{aligned}\)

\(LHS = RHS\)

Hence, it is proved that \(A \cup B = B\).

(b) Showing that \(A \cap B = A\).

It is given that \(A \subseteq B\). Thus,

\(A \subseteq B = \left\{ {x|x \in A \Rightarrow x \in B} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{a}}\;{\rm{subset}}\)

Solve the left hand side of set \(A \cup B\) as follows:

\(\begin{aligned}LHS &= A \cap B\\ &= \left\{ {x|x \in A \wedge x \in B} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{union}}\\ &= \left\{ {x|x \in A \wedge x \in A} \right\}\;{\rm{by}}\;{\rm{definition}}\;{\rm{of}}\;{\rm{a}}\;{\rm{subset}}\\ &= A\end{aligned}\)

\(LHS = RHS\)

Hence, it is proved that \(A \cap B = A\).