Question

Let A and B be subsets of a universal set U. Show that \(A \subseteq B\) if and only if \(\overline B \subseteq \overline A \).

Short answer

Thus, we have to prove \(A \subseteq B\) if and only if\(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \).

Step-by-step solution

Given

Let A and B be subsets of a universal set U.

solution

Here we solve our question in two-part (i) If part and (ii) only if part.

Here (i) if part

Let\(A \subseteq B\).

We have to show that

\(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \).

Let \(x \in \mathop B\limits^\_ \)

\( \Rightarrow x \notin {\bf{B}}\)

\( \Rightarrow x \notin A\)(\(A \subseteq B\))

\( \Rightarrow x \in \mathop A\limits^\_ \)

Hence, \(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \).

Now here we solve (ii)

Let\(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \).

We have to show that\(A \subseteq B\).

Let \(x \in A\)

\( \Rightarrow x \notin \mathop A\limits^\_ \)

\( \Rightarrow x \notin \mathop B\limits^\_ \)(\(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \))

\( \Rightarrow x \in B\)

Hence, \(A \subseteq B\)

Thus we get,

\(A \subseteq B\)iff\(\mathop B\limits^\_ \subseteq \mathop A\limits^\_ \).

Hence, this is proved.