Question

Consider the given statement and verify whether the statement is true or false
\[\{x:x\in W,x+5=5\}=\varphi \]

Answer 1

Hint:From the given condition, check if they contain any elements or not. In a null set there won’t be any value that will satisfy the condition. If there are values that satisfy the condition then it is not a null set.

Complete step-by-step answer:
The empty set is the unique set having no elements, its size or cardinality, which is the count of elements in the set is zero. We can ensure that the empty set exists by including an axiom of empty set. Now the empty set can be also referred to as a null set. However null set is a distinct notion within the context of measure theory.
Null or empty set can be also described as a set of measure zero. The common notations for the empty set include “{ }” and $\phi $ .
Now, we have been given, $\{x:x\in W,x+5=5\}=\phi $
Here $x$ belongs to a whole number:
Let us solve $\begin{align}
  & x+5=5, \\
 & x=5-5=0 \\
 & x=0 \\
\end{align}$
Thus, we get an element zero, which is a whole number. Thus, the element of the set $\{x:x\in W,x+5=5\}=\{0\}$ which is not a null set because it contains one element.
Hence the statement is false.

Note: For statements like this can never have a null set. Solution of a linear equation like $x+5=5,$ will always have 0 value for $x$ . Irrespective of what operation is done. The same is the case for quadratic equations or any other polynomial.