UPSKILL MATH PLUS

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Consider 'the set of negative numbers in natural number system'.

This collection is well defined because we have to pick negative numbers in the natural number system.

The fact is that we don't have any negative numbers in the natural number system.

In the natural number system, we have only positive numbers.

So there is no element in this collection.

A set that does not contain any element is called empty or null or void set. It is denoted by $\varnothing \phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}\left\{\right\}$.

Example:

- \(A=\{x:x\) is a perfect square number between \(18\) and \(24\}\). There is not a perfect square number between \(18\) and \(24\). So the set \(A\) is an empty set.

- The set of natural numbers less than \(1\). We know that the natural number starts with \(1\). There is no natural number less than \(1\). Thus the set is empty.
- The set of all integers between \(5\) and \(6\). Integers are whole numbers. It cannot be a fractional number. Thus, there is no integer between \(5\) and \(6\). Therefore, the set is empty.
- The set of all even prime number greater than \(5\). The only even prime is \(2\). There is no even prime number greater than \(5\). Thus the set is empty.

A set that contains only one element is called a singleton set.

Example:

- The set of whole numbers between \(10\) and \(12\). Here the set $A=\; \{11\}$.
- The set $B=\; \{a\}$ be the singleton set.
- The set of natural number which is neither prime nor composite. The number \(1\) is neither prime nor composite. But number \(1\) is a natural number. Thus the set $C=\{1\}$.
- The set of integer which is neither positive nor negative. The integer \(0\) is neither positive nor negative. Thus the set $D=\; \{0\}$.
- \(E=\{x | x\) is a prime number less than \(3\}\). We know that the number \(2\) is the only prime less than 3. Thus the set $E=\; \{2\}$.