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

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 $A \subset B$.