Question

State whether the following statement is true or false:
The set {x: x + 8 = 8} is the null set

Answer 1

Hint:In this question, from the given relation, find the value of x. If none of the value of x satisfies the relation, then the given set is null otherwise it is not.

Complete step-by-step answer:
In this question, we have to tell if set {x: x + 8 = 8} is a null set or not. Before proceeding with this question, let us understand a few terms:
Sets: A set is a well defined collection of objects. Eg: {x: 1, 2, 3, 4, ….} is a set of natural numbers or if set \[\left\{ x:{{x}^{2}}-1=0 \right\}\] then {x: 1, – 1} etc.
Null set: The null set is a set that contains no elements. We can also refer to the null set as an empty set or as the set that does not contain any elements. We use \[x\in \phi \] or x: { } to show that x is a null set or empty set. For example: if A : { x: 9 < x < 10, x is a natural number} is a null set because there are no natural numbers between 9 and 10.
Now, let us consider our question. Let us consider the given set
A: {x: x + 8 = 8}
By solving the given relation, that is:
x + 8 = 8
We get, x = 8 – 8 = 0
So, x = 0 satisfies this relation, given that set A would have 1 element and that is 0. So, we can write set A as A: {0}. So, the given set is not null as the set is not empty because it contains an element and that is 0.
So, our answer is false.

Note: In these types of questions, students often make this mistake of considering set as null set when it contains an element 0 which is wrong. Students must note that 0 is also an element of a set and when 0 is in the set, the set is not empty. Also, students must understand the difference between 0 as an element of the set and the total number of elements in the set as 0. In the latter case, the set is the null set.