Question

Which of the following is a true statement?

1. \left\\{ a \right\\}\in \left\\{ a,b,c \right\\}
2. \left\\{ a \right\\}\subseteq \left\\{ a,b,c \right\\}
3. \phi \in \left\\{ a,b,c \right\\}
4. None of these

Answer 1

Here in this question we have been asked to evaluate the statements in the options and mark the correct ones. Considering that $a,b,c$ are three different elements. \left\\{ a,b,c \right\\} is a set consisting of those three different elements . We will verify the given statements.

Complete step-by-step solution:
Now considering from the question we can say that $a,b,c$ are three different elements and \left\\{ a,b,c \right\\} is a set consisting of those three different elements.
Now we can say that a\in \left\\{ a,b,c \right\\} , b\in \left\\{ a,b,c \right\\} and c\in \left\\{ a,b,c \right\\} this implies that the element $a,b,c$ belongs to the set \left\\{ a,b,c \right\\} .
Hence we can conclude that the statements \left\\{ a \right\\}\in \left\\{ a,b,c \right\\} and \phi \in \left\\{ a,b,c \right\\} are false because $\phi$ is not present in the set \left\\{ a,b,c \right\\} and the set \left\\{ a \right\\} does not belongs to the set \left\\{ a,b,c \right\\} .
From the basic concepts of sets we know that the set $B$ is said to be a subset of the set $A$ if and only if every element in the set $B$ belongs to set $A$ .
Here we can say that every element in the set \left\\{ a \right\\} belongs to the set \left\\{ a,b,c \right\\} then we can say that the set \left\\{ a \right\\} is that subset of \left\\{ a,b,c \right\\} which is mathematically represented as \left\\{ a \right\\}\subseteq \left\\{ a,b,c \right\\} .
Here we have one valid statement in the concept saying that “Null set is the subset of every set.”
Here by using the above statement we can say that \phi \subseteq \left\\{ a,b,c \right\\} .
Therefore we can conclude that only statement 2 is true.
Hence we will mark the option “2” as correct.

Note: While answering questions of this type we should be sure with the notations that we are using. Here you can observe that there is a difference between saying the element belongs to the set and the set is a subset of the given set. This difference is only identified by the notation someone can get confused between the first two statements and can mark both as correct or only the first one as correct if both the cases are wrong.