Question
Which of the following is true ?
A.
$$a\, \in \,\left\{ {\left\{ a \right\},\,b} \right\}$$
B.
$$\left\{ {b,\,c} \right\}\, \subset \left\{ {a,\,\left\{ {b,\,c} \right\}} \right\}$$
C.
$$\left\{ {a,\,b} \right\}\, \subset \left\{ {a,\,\left\{ {b,\,c} \right\}} \right\}$$
D.
none of these
Answer :
none of these
Solution :
$$a$$ is not an element of $$\left\{ {\left\{ a \right\},\,b} \right\}$$
$$\therefore \,a\, \notin \,\left\{ {\left\{ a \right\},\,b} \right\}$$
$$\left\{ {b,\,c} \right\}$$ is the element of $$\left\{ {a,\,\left\{ {b,\,c} \right\}} \right\}$$
$$\eqalign{
& \therefore \,\left\{ {b,\,c} \right\}\, \in \,\left\{ {a,\,\left\{ {b,\,c} \right\}} \right\} \cr
& b\, \in \,\left\{ {a,\,b} \right\}{\text{ but }}b \notin \,\left\{ {a,\,\left\{ {b,\,c} \right\}} \right\} \cr
& \therefore \,\left\{ {a,\,b} \right\}\, \not\subset \,\left\{ {a,\,\left\{ {b,\,c} \right\}} \right\} \cr} $$