Question

Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Verify whether the following statement is true or false. Why?
1)\[\varphi \subset A\]

Answer 1

Hint: The symbol ‘\[\subset \]’ represents a proper or strict subset so first assess the statement by comparing the elements and they write true / false.

Complete step-by-step answer:

In the question we are given a set A such that it represents as {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Further a statement is written that \[\varphi \subset A\] and we have to say that it is true or false.

At first we brief ourselves by understanding what sets are.

In mathematics, sets 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 number 2, 4, 6 are distinct and considered separately, but they are considered collectively they form a single set of size three written as {2, 6, 4}.

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

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

Here A is {{1, 2, 3}, {4, 5}, {6, 7, 8}}. So, its subsets will be { }, {1, 2, 3}, {4, 5}, {6, 7, 8}, {{1, 2, 3}, {4, 5}}, {{4, 5}, {6, 7, 8}}, {{1, 2, 3}, {6, 7, 8}}, {{1, 2, 3}, {4, 5}, {6, 7, 8}}.

For proper subsets we will omit { } from total subsets.

Now we can see that \[\varphi \] is not any of the 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.