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In set theory, understanding what a subset is lies at the heart of grasping more complex concepts. When we say that set A is a subset of set B, denoted as \(A \subseteq B\), it means every element in set A is also present in set B. Imagine you have a basket of fruits. If the set A consists of only apples and set B contains all the fruits in the basket, then A would be a subset of B. This is because every apple (an element of A) is also one of the fruits in the basket (an element of B).
Some important properties of subsets include:

• Every set is a subset of itself, so for any set A, \(A \subseteq A\).
• The empty set is a subset of every set, so for any set B, \( \emptyset \subseteq B \).

Understanding subsets is vital as it forms the base for relating sets through their members.

Cardinality is a term that represents the number of elements in a set. It helps us compare the sizes of different sets. If set A = {1, 2, 3}, the cardinality of A, denoted as \(|A|\), is 3 because it has 3 elements. Similarly, if B = {1, 2, 3, 4, 5}, then \(|B|\) is 5.
Some key points about cardinality are:

• The cardinality of the empty set \( \emptyset \) is 0 since it has no elements.
• If two sets have the same number of elements, they have the same cardinality. For instance, if set C = {a, b, c}, then \( |C| =|A|=3 \) because both sets have three elements.

When we talk about unequal cardinality, we say one set has less than or equal to the number of elements compared to another. This is fundamental when we relate set inclusion to cardinality.

Set inclusion connects the concepts of subsets and cardinality. If \(A \subseteq B\), it translates to \(|A| \leq |B|\). This is because every element in A is already counted within B, and therefore, the total number of elements in A cannot exceed that in B. To visualize this, think of a classroom (set B) with all students. If set A represents a subset of students who wear glasses, this group (A) is part of the larger group (B), making their count smaller or equal.
Also, in mathematical proof, we often move from showing set inclusion to proving the inequality of their cardinal numbers. This is straightforward:

• If \(A \subseteq B\), then every element of A appears in B.
• This implies that counting all unique elements in A will always be less than or equal to the count of elements in B.

This simple relation is the cornerstone for numerous proofs and applications in set theory.