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A part of the set A is called a subset of the set A.

For instance, the set of equilateral triangles E is a subset of the set of triangles in T.

Note that each element of E is an element of T. In symbols, we express this fact as E T. We read it as "E is a subset of T" or "E is contained in T". The definition reads as follows


A set S is a subset of a set R, if every element of S is an element of R. That is, S R if whenever x S then x R.
We now introduce a symbol "" which means "implies". We now rewrite the above definition using this symbol.
S R if x S x R.

We read this as "S is a subset of R if x is an element of S implies x is also an element of R."

From this diagram, we can say that A is a subset of B.

Illustration 1
The set A = {a, b, c} is a subset of the set B = {b, c, d, a}, because each element a, b and c of the set A is also an element of the set B.

Illustration 2
Let E = {2, 4, 6, 8, ...}, and let P = {2n}. Then P E, i.e. P is contained in E.

Illustration 3
Let A = {x | x is a student of your class} and B = {x | x is a student of your school}, then A B.

Illustration 4
Let A = {1, 3, 5} and B = {1, 3, 5}. Since, each element 1, 3 and 5 of A is also an element of B, we have A B. Note, in particular, that A = B.

Since every fruit in the set A is present in set B, we conclude that A = B.

In both the examples, we saw that if A = B, then A B. In general, A A for every set A, Also, note that the empty set φ, which contains no element is a subset of every set.

Using the definition of subset, the definition of equality of two sets can be stated as follows:
Two sets A and B are equal, i.e. A = B, if and only if A B and B A.

This can be written as A B, B ⊂ A ⇔ A = B, where '' is a symbol for two way implications, and is usually read as if and only if.

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