Question

Which one of the following is an example of a non-empty set?
A) Set of all prime numbers
B) {$x:{x^2} - 2 = 0$ and $x$ is rational}
C) {$x:x$ is a natural number,$x < 8$ and simultaneously $x > 12$}
D) {$x:x$ is a point common to any two parallel lines}

Answer 1

We are given a few examples of sets out of which we have to find out which examples are non-empty sets. Any grouping of elements which satisfies the properties of a set in which has at least one element with an example of non-empty set. The set S$= \\{ 1,4,5\\}$ is a non-empty set.
To find this we have to apply different concepts to different examples. For example two Parallel Lines do not have any common point no natural number be small and greater to a certain number at the same time. By applying such concepts we have to find out which form a set that has some elements or a non-empty set

Complete step-by-step answer:
Step1: Our first example given is a set of all even prime numbers. Prime numbers are the numbers that have only two common factors that are one and themselves but even numbers have a common factor $2$also. So the only even prime number is $2$ whose factors are $1$ and $2$ itself.
So set is $\\{ 2\\}$
As this set contains elements it is a non-empty set.
Step2: Our second example is {$x:{x^2} - 2 = 0$ and $x$ is rational}. The number whose difference with $2$ is equal to zero can only be possible when the number is $2$ itself. Here ${x^2}$ must be $2$. Hence $x$ must be equal to $\sqrt 2$ but $\sqrt 2$ is irrational. Here $x$ is required to be rational. Hence this is not possible. Set is empty.
Step3: In this example we are given {$x:x$ is a natural number, $x < 8$ and simultaneously $x > 12$}. We are given that $x$ must be less than $8$ and greater than $12$ which is impossible at the same time. Hence the set is empty.
Step4: We are given the next example {$x:x$ is a point common to any parallel lines}. Two Parallel Lines cannot intersect each other and when they don't intersect then there will be no common point. Hence this set is also empty

Hence, option (A) is the correct Answer.

Note: A set 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. A set may be denoted by placing its objects between a pair of curly braces.