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# 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 $N - A = \left\{ {n \in N,n \notin \left. A \right\}} \right.$ isA. Empty SetB. Of cardinality 2 C. A finite set of cardinality greater than 2D. A singleton set

Last updated date: 25th Jul 2024
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Hint: 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,$$n$is a multiple of some prime $\left. {p \in P} \right\},$
$N - A = \left\{ {n \in N,n \notin \left. A \right\}} \right.$
Now,
$N$denotes the set of natural numbers
$\Rightarrow N = \left\{ {1,2,3,4,5,6,7,...} \right\}$
$A = \left\{ {n:n} \right. \in N,$$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.