Question

If \[X = \left\{ {a,\left\{ {b,c} \right\},d} \right\}\], which of the following is a subset of \[X\] \[?\]
A. \[\left\{ {a,b} \right\}\]
B. \[\left\{ {b,c} \right\}\]
C. \[\left\{ {c,d} \right\}\]
D. \[\left\{ {a,d} \right\}\]

Answer 1

Hint: First we have to know that a set \[S\] is said to be a subset of another set \[A\] if all the elements of the set \[S\]are the elements of the set \[A\]. Which is denoted by \[S \subseteq A\]. Hence, every subset of a set is made by the elements of that set.

Complete answer:
Given \[X = \left\{ {a,\left\{ {b,c} \right\},d} \right\}\] is a set. Then the elements of \[X\]are \[a,\left\{ {b,c} \right\},d\].
Hence the subset of \[X\] are formed using elements \[a,\left\{ {b,c} \right\},d\]
A. \[\left\{ {a,b} \right\}\]
Since \[a \in X\]but \[b \notin X\].
Then a set \[\left\{ {a,b} \right\}\]is not a subset of \[X\].
B. \[\left\{ {b,c} \right\}\]
Since \[\left\{ {b,c} \right\} \in X\], which implies that \[\left\{ {b,c} \right\}\]is an element of \[X\].
Then, \[\left\{ {b,c} \right\}\]is not a subset of \[X\].

C. \[\left\{ {c,d} \right\}\]
Since \[d \in X\]but \[c \notin X\].
Then a set \[\left\{ {c,d} \right\}\]is not a subset of \[X\].

D. \[\left\{ {a,d} \right\}\]
Since \[a,d \in X\].
Then a set \[\left\{ {a,d} \right\}\]is a subset of \[X\].
Hence the correct option is (D) \[\left\{ {a,d} \right\}\].
Therefore, the correct option is D

Note: Note that a set is a well-defined collection of objects. Also note that the subsets of a set are classified into proper and in-proper subsets. The null set (a set that contains no elements) and set itself are the in-proper subsets and other subsets are proper subsets of a given set i.e., If a subset \[A\] is a proper subset of a set \[B\] then it denoted by \[A \subset B\].