22.

Consider the proper subsets of {1, 2, 3, 4). How many of these subsets are superset of the set {3} ?

• 5

• 6

• 7

• 8

Let A = {1, 2, 3, 4}

Then, Subsets of A,

$$\begin{gathered} \mathrm{B} = \mathrm{\phi }, \left\{1\right\}, \left\{2\right\}, \left\{3\right\}, \left\{4\right\}, \left\{1, 2\right\}, \left\{2, 3\right\}, \left\{1, 3\right\}, \left\{2, 4\right\}, \left\{1, 4\right\}, \left\{3, 4\right\}, \left\{1, 2, 3\right\}, \left\{1, 2, 4\right\}, \left\{2, 3, 4\right\}, \left\{1,3, 4\right\}, \left\{1, 2, 3, 4\right\} \\ \mathrm{And} \mathrm{we} \mathrm{know} \mathrm{that} \mathrm{a} \mathrm{subset} \mathrm{B} \mathrm{of} \mathrm{a} \mathrm{set} \mathrm{A} \mathrm{is} \mathrm{called} \mathrm{a} \mathrm{proper} \mathrm{subset} \mathrm{of} \mathrm{A} \\ \mathrm{if} \mathrm{A} \ne \mathrm{B} \\ \mathrm{Then} \mathrm{proper} \mathrm{subset} \mathrm{of} \mathrm{A} \\ = \left\{1\right\}, \left\{2\right\}, \left\{3\right\}, \left\{4\right\}, \left\{1, 2\right\}, \left\{2, 3\right\}, \left\{1, 3\right\}, \left\{2, 4\right\}, \left\{1, 4\right\}, \left\{3, 4\right\}, \left\{1, 2, 3\right\}, \left\{1, 2, 4\right\}, \left\{2, 3, 4\right\}, \left\{1,3, 4\right\} \\ \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{whenever} \mathrm{a} \mathrm{set} \mathrm{B} \mathrm{is} \mathrm{a} \mathrm{subset} \mathrm{of} \mathrm{set} \mathrm{A}, \mathrm{we} \mathrm{say} \mathrm{that} \mathrm{A} \mathrm{is} \mathrm{a} \mathrm{superset} \mathrm{of} \mathrm{B} \mathrm{or} \mathrm{A}⊒ \mathrm{B} \\ \mathrm{So},\mathrm{super} \mathrm{set} \mathrm{of} \left\{3\right\} \mathrm{in} \mathrm{subset} \mathrm{of} \mathrm{B} \\ = \left\{1\right\}, \left\{2\right\}, \left\{4\right\}, \left\{1, 2\right\}, \left\{2, 4\right\}, \left\{1, 4\right\}, \left\{1, 2, 4\right\} \end{gathered}$$