Question

Let $A=\left\{ a,b,\left\{ c,d \right\},e \right\}$ . Verify whether the following statement is true or false. Why?
(i) $\varphi \subset A$

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 / false.

Complete step-by-step answer:
In the given question, we are given a set A such that it represents $\left\{ a,b,\left\{ c,d \right\},e \right\}$. Further a statement is written $\varphi \subset A$ and we have to say that is true or false.
At first, we briefly understand what is set.
In mathematics sets is 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 number 2, 4, 6 are distinct and considered separately, but they are considered collectively then for mn single set of size three written as $\left\{ 2,4,6 \right\}$ which could also be written as $\left\{ 2,6,4 \right\}$ .
There are various symbols used in sets and each has a different meaning. Here in the statement the 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 find strict or proper subsets of A. So, here subsets of A is
$\left\{ {} \right\}$ , $\left\{ a \right\}$ , $\left\{ b \right\}$ , $\left\{ \left\{ c,d \right\} \right\}$ , $\left\{ e \right\}$ , $\left\{ a,b \right\}$ , $\left\{ a,\left\{ c,d \right\} \right\}$ , $\left\{ a,e \right\}$ , $\left\{ b,\left\{ c,d \right\} \right\}$ , $\left\{ b,e \right\}$ , $\left\{ \left\{ c,d \right\},e \right\}$ , $\left\{ a,b,\left\{ c,d \right\} \right\}$ , $\left\{ a,b,e \right\}$ , $\left\{ b,\left\{ c,d \right\},e \right\}$ , $\left\{ a,\left\{ c,d \right\},e \right\}$ , $\left\{ a,b,\left\{ c,d \right\},e \right\}$ .
For subsets, we will omit $\left\{ {} \right\}$ from total subsets.
Now, we can see that $\varphi $ does not have any subset of A. So, the given statement is not true.
Hence, the statement is false.

Note: Students generally have confusion between these symbols as they are so much used in the sets just like confusion between $\in $ and $\subset $ , where former represents set membership and latter one represents one subset of another.