Let A $$={1$$, $$2$$,$$3$$, $$4}$$, B $$=$$ $${2$$, $$4$$, $$6}$$. Then, the number of sets C such that A∩B⊆C⊆A∪B is

we have,A∩$$B={2$$,$$4}$$⇒$$n(A$$∩$$B)=2A$$∪$$B={1$$,$$2$$,$$3$$,$$4$$,$$6}$$⇒$$n(A$$∪$$B)=5as$$ A∩B⊆C⊆A∪B Set C contains the number of elements $$=25$$−$$2=23=8$$

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Question

Let A ={1, 2,3, 4}, B = {2, 4, 6}. Then, the number of sets C such that $A \cap B \subseteq C \subseteq A \cup B$ is

Moderate

Solution

we have,
$\begin{aligned} A \cap B & = {2, 4} \\ \Rightarrow n (A \cap B) & = 2 \\ A \cup B & = {1, 2, 3, 4, 6} \\ \Rightarrow n (A \cup B) & = 5 \\ as A \cap B \subseteq C \subseteq A \cup B \end{aligned}$
Set C contains the number of elements
$= 2^{5 - 2} = 2^{3} = 8$

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