Question

Let \[A=\left\{ \varphi ,\left\{ \varphi \right\},1,\left\{ 1,\varphi \right\},2 \right\}\]. Verify whether the following statement is true or false. Why?
\[\left\{ \varphi \right\}\in A\]

Answer 1

Hint: The symbol ‘\[\in \]’ represents set membership so first assess the statement by comparing the elements and then write true or false.

Complete step-by-step answer:

In the question we are given a set A that is represented as {\[\varphi \], {\[\varphi \]}, 1, {1, \[\varphi \]}, 2}. Further a statement is written that \[\left\{ \varphi \right\}\in A\] and we have to say that it is true or false.

At first we will briefly understand what is a set.

In mathematics seta are a well defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the numbers 2, 4, 6 are distinct and considered separately, but they are considered collectively they form a single set of size three written as {2, 4, 6} which could also be written as {2, 6, 4}.

There are various symbols used in sets and each has a different meaning. Here, in the statement a symbol ‘\[\in \]’ is given. This symbol’s name is an element of or belongs to which symbolizes set memberships for example if C = {1, 2, 3} then, we can say 1 \[\in \] C.

In ‘A’ elements are \[\varphi \], {\[\varphi \]}, 1, {1, \[\varphi \]} and 2, in the question we are asked whether \[\left\{ \varphi \right\}\in A\] is true or false. As we see that \[\left\{ \varphi \right\}\] is one of the elements of A.

Hence, the given statement is true.

Note: Students generally confuse between \[\varphi \] and {\[\varphi \]} as they sometimes they are same but they are not as \[\varphi \] is not considered as set {\[\varphi \]} is considered as singleton set or set with one element.