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?

1) \[\left\{ \varphi ,\left\{ \varphi \right\},\left\{ 1,\varphi \right\} \right\}\subset A\]

Answer 1

Hint: The symbol ‘ $\subset $ ’ represents a proper or strict subset, 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 represents as \[\left\{ \varphi ,\left\{ \varphi \right\},1,\left\{ 1,\varphi \right\},2 \right\}\] . Further, a statement is written that \[\left\{ \varphi ,\left\{ \varphi \right\},\left\{ 1,\varphi \right\} \right\}\subset A\] and we have to say that it is true or false.

At first, we briefly understand what is set.

In mathematics sets are well defined collections 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 collectively which for m, n single set of size three written as \[\left\{ 2,4,6 \right\}\] which can be also written as \[\left\{ 2,6,4 \right\}\] .

There are various symbols used in sets and each has a different meaning. Here, in the statement a symbol ‘ $\subset $ ’ is given. This symbol’s name is proper subset or strict subset such as for example \[\left\{ 9,14 \right\}\subset \left\{ 9,14,28 \right\}\]

Let us first find the strict or proper subset of A.

Here A is \[\left\{ \varphi ,\left\{ \varphi \right\},1,\left\{ 1,\varphi \right\},2 \right\}\] so, its subsets will be \[\left\{ {} \right\}\] , \[\left\{ \varphi \right\}\] , \[\left\{ \left\{ \varphi \right\} \right\}\] , \[\left\{ 1 \right\}\] , \[\left\{ \left\{ 1,\varphi \right\} \right\}\] , \[\left\{ 2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\} \right\}\] , \[\left\{ \varphi ,1 \right\}\] , \[\left\{ \varphi ,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ \varphi ,2 \right\}\] , \[\left\{ \left\{ \varphi \right\},1 \right\}\] , \[\left\{ \varphi ,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ \left\{ \varphi \right\},2 \right\}\] , \[\left\{ 1,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ 1,2 \right\}\] , \[\left\{ \left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},1 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},2 \right\}\] , \[\left\{ \varphi ,1,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ \varphi ,1,2 \right\}\] , \[\left\{ \varphi ,\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \left\{ \varphi \right\},1,2 \right\}\] , \[\left\{ \left\{ \varphi \right\},\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \left\{ \varphi \right\},1,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ 1,\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},1,\left\{ 1,\varphi \right\} \right\}\] , \[\left\{ \varphi ,1,\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},1,2 \right\}\] , \[\left\{ \left\{ \varphi \right\},1,\left\{ 1,\varphi \right\},2 \right\}\] , \[\left\{ \varphi ,\left\{ \varphi \right\},1,\left\{ 1,\varphi \right\},2 \right\}\] .


For a proper subset we will omit \[\left\{ {} \right\}\] from the total subsets.

We can see that \[\left\{ \varphi ,\left\{ \varphi \right\},\left\{ 1,\varphi \right\} \right\}\] is one of subset of A so, the given statement is correct.

Hence, the statement is true.

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