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.