Question

Is B a subset of C when

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathit{B}\mathbf{=}\cancel{\mathbf{c}}\mathbf{}\mathbf{and}\mathbf{}\mathit{C}\mathbf{=}\mathbf{¤} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{}\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{all}\mathbf{}\mathbf{soluation}\mathbf{}\mathbf{of}\mathbf{}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{5}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{and}\mathbf{}\mathit{C}\mathbf{=}\cancel{\mathbf{c}}\mathbf{?} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{}\mathit{B}\mathbf{=}\mathbf{\{}\mathbf{a}\mathbf{,}\mathbf{b}\mathbf{,}\mathbf{7}\mathbf{,}\mathbf{9}\mathbf{,}\mathbf{11}\mathbf{,}\mathbf{-}\mathbf{6}\mathbf{\}}\mathbf{}\mathbf{and}\mathbf{}\mathit{C}\mathbf{=} \end{gathered}$$

Short answer

(a) B is a subset of C .

(b) Bis not a subset of C .

(c) B is not a subset of C .

Step-by-step solution

(a) Step 1: When B=c and C=¤

As given, that $B=\cancel{c}$means the set of integers and $C=\mathbf{¤}$means the set of rational numbers.

From the definition, every integer is a rational number but a rational number need not be an integer and subset means a set of which all the elements are contained in another set.

Hence, B is a subset of C.

(b) Step 2: When B = all soluation of x2+2x-5=0 and C=c?

As it is given that $B=\mathrm{all}\mathrm{soluation}\mathrm{of}{x}^2+2x-5=0\mathrm{and}C=\cancel{\mathrm{c}}?$and .

Thus, factorise the given equation as follows:

$$\begin{gathered} x^2+2x-5=0 \\ x=\frac{-2\pm \sqrt{4-4.1.(-5})}{2} \\ =\frac{-2\pm \sqrt{24}}{2} \\ =-1\pm \sqrt{6} \end{gathered}$$

Now, $-1+\sqrt{6}\notin \cancel{c}$,therefore is not a subset of C.

Hence, B is not a subset of C.

(c) Step 3: When B={a,b,7,9,11,-6} and C=¤

As it is given that $B=\{\mathrm{a},\mathrm{b},7,9,11,-6\}\mathrm{and}C=$$¤$.

Now, every integer is a rational number but a rational number need not be an integer. Therefore, B is not a subset of C.