Give an example of a set S such that (a) S ⊆𝒫(𝐍) (b) S ∈𝒫(𝐍) (c) S ⊆𝒫(𝐍) and |S|=5 (d) S ∈𝒫(𝐍) a

step 1 Hello, so for part a here to determine the number of subsets of cardinality four from the set s

Give an example of a set S such that (a) S ⊆𝒫(𝐍) (b) S ∈𝒫(𝐍)...

Give an example of a set $S$ such that
(a) $S \subseteq \mathcal{P}(\mathbf{N})$
(b) $S \in \mathcal{P}(\mathbf{N})$
(c) $S \subseteq \mathcal{P}(\mathbf{N})$ and $|S|=5$
(d) $S \in \mathcal{P}(\mathbf{N})$ and $|S|=5$

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Key Concepts

Element-of Relation

The element-of relation, denoted by ?, signifies membership. When we say S ? P(N), it means that S is one of the subsets contained in the power set of the natural numbers. This is different from being a subset; a set can be an element of a power set if it is one of its members.

Cardinality

Cardinality is the concept that gives the size, or the number of elements, in a set. In the context of the problem, specifying |S| = 5 means that there are exactly five elements in the set S. This concept is crucial, especially when constraining sets to have a particular finite size while still adhering to other conditions such as being a subset of or an element of a power set.

Power Set

The power set of a set, denoted by P(N) in this context, is the set of all possible subsets of the given set (here, the natural numbers). This concept is fundamental in set theory as it demonstrates how many ways a set can be partitioned into its potential subsets, including the empty set and the set itself.

Subset Relation

The subset relation, denoted by ?, is a key concept where one set S is considered a subset of another set T if every element of S is also an element of T. In this problem, it's important to recognize that stating S ? P(N) means each element of S is itself a subset of the natural numbers.

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