Question

Let $N$ be the set of natural numbers and $P$ be the set of Prime integers in $N$. If A = \left\\{ {n:n} \right. \in N,$$$$n is a multiple of some prime \left. {p \in P} \right\\}, then N - A = \left\\{ {n \in N,n \notin \left. A \right\\}} \right. is
A. Empty Set
B. Of cardinality 2
C. A finite set of cardinality greater than 2
D. A singleton set

Answer 1

Observing the sets carefully, we get to know that, $N$ is a set of all natural numbers, and $A$ is a set containing all the prime numbers and all the multiples of all prime numbers which implies that the set $A$ contains all the natural numbers except $1$. And now, if we subtract set $A$ from set $N$,
We get only one element left in the resulting set i.e. $1$

Complete step by step answer:
Given, $N$represents the set of natural numbers, and
$P$ denotes the set of Prime integers in$N$.
A = \left\\{ {n:n} \right. \in N,$$$$nis a multiple of some prime \left. {p \in P} \right\\},
N - A = \left\\{ {n \in N,n \notin \left. A \right\\}} \right.
Now,
$N$denotes the set of natural numbers
\Rightarrow N = \left\\{ {1,2,3,4,5,6,7,...} \right\\}
A = \left\\{ {n:n} \right. \in N,$$$$n is a multiple of some prime \left. {p \in P} \right\\}
\Rightarrow A = \left\\{ {n:n \in N,n = kp,p \in P,k \in N} \right\\},
\Rightarrow A = \left\\{ {2,3,4,5,6,7,.......} \right\\}
N - A = \left\\{ {n \in N,n \notin \left. A \right\\}} \right.
Hence,N - A = \left\\{ 1 \right\\}, which is a singleton set.

So, the correct answer is “Option D”.

Note: A set is called an Empty set if it does not contain any element. It is denoted by \left\\{ {} \right\\} or $\phi$.
A set of cardinality 2 means a set having two elements such as A = \left\\{ {a,b} \right\\}is a set having cardinality 2.
A finite set having cardinality greater than 2 means a set which has a definite number of elements but more than two elements such as $A =${North, West, South, East} .
If a set $A$ has only one element, then set $A$ is called a singleton set. Like \left\\{ a \right\\} is a singleton set.