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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 \[N - A = \left\{ {n \in N,n \notin \left. A \right\}} \right.\] is
A. Empty Set
B. Of cardinality 2
C. A finite set of cardinality greater than 2
D. 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 \[\left. {p \in P} \right\},\]
\[N - A = \left\{ {n \in N,n \notin \left. A \right\}} \right.\]
\[N\]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.