**Subsets**

Consider the sets : X = set of all students in your school, Y = set of all students in your class.

We note that every element of Y is also an element of X; we say that Y is a subset of X. The fact that Y is subset of X is expressed in symbols as Y ⊂ X. The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.

**Definition 4** A set A is said to be a **subset** of a set B if every element of A is also an element of B.

In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol “⇒” which means implies. Using this symbol, we can write the **definition of subset** as follows:

A ⊂ B if a ∈ A ⇒ a ∈ B

We read the above statement as “A is a subset of B if a is an element of A implies that a is also an element of B”. If A is not a subset of B, we write A ⊄ B.

We may note that for A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A. If it so happens that every element of B is also in A, then we shall also have B ⊂ A. In this case, A and B are the same sets so that we have A ⊂ B and B ⊂ A ⇔ A = B, where “⇔” is a symbol for two way implications, and is usually read as if and only if (briefly written as “iff”).

It follows from the above definition that every set A is a subset of itself, i.e., A ⊂ A. Since the empty set φ has no elements, we agree to say that φ is a subset of every set. We now consider some examples :

(i) The set **Q** of rational numbers is a subset of the set **R** of real numbes, and we write **Q** ⊂ R.

(ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56, then B is a subset of A and we write B ⊂ A.

(iii) Let A = {1, 3, 5} and B = {x : x is an odd natural number less than 6}. Then A ⊂ B and B ⊂ A and hence A = B.

(iv) Let A = { a, e, i, o, u} and B = { a, b, c, d}. Then A is not a subset of B, also B is not a subset of A.

Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a **proper subset** of B and B is called **superset**of A. For example,

A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.

If a set A has only one element, we call it a **singleton set**. Thus,{ a } is a singleton set.

**Example 9** Consider the sets

φ, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.

Insert the symbol ⊂ or ⊄ between each of the following pair of sets:

(i) φ. . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

**Solution**

(i) φ ⊂ B as φ is a subset of every set.

(ii) A ⊄ B as 3 ∈ A and 3 ∉ B

(iii) A ⊂ C as 1, 3 ∈ A also belongs to C

(iv) B ⊂ C as each element of B is also an element of C.

**Example 10** Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a subset of B ? No. (Why?). Is B a subset of A? No. (Why?)

**Example 11** Let A, B and C be three sets. If A ∈B and B ⊂ C, is it true that A ⊂ C?. If not, give an example.

**Solution** No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ∈ B as A = {1} and B ⊂ C. But A ⊄ C as 1 ∈ A and 1 ∉ C.

Note that an element of a set can never be a subset of itself.

_{Subsets of set of real numbers}

There are many important subsets of **R**. We give below the names of some of these subsets.

The set of natural numbers N = {1, 2, 3, 4, 5, . . .}

The set of integers Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}

The set of rational numbers Q = { x : x = p /q , p, q ∈ Z and q ≠ 0}

which is read “**Q** is the set of all numbers x such that x equals the quotient p/q , where p and q are integers and q is not zero”. Members of **Q** include –5 (which can be expressed as -5/1), 5/7, 3 ½ (which can be expressed as 7/2) and -11/3 .

The set of irrational numbers, denoted by **T**, is composed of all other real numbers. Thus **T** = {x : x ∈ **R**and x ∉ **Q**} = **R** – **Q**., i.e., all real numbers that are not rational. Members of **T** include √2 , √5 and π .

Some of the obvious relations among these subsets are:

**N** ⊂ **Z** ⊂ **Q**, **Q** ⊂ **R**, **T** ⊂ **R**, **N** ⊂ **T**.

_{Intervals as subsets of R}

Let a, b ∈ **R** and a < b. Then the set of real numbers { y : a < y < b} is called an open interval and is denoted by (a, b). All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval.

The interval which contains the end points also is called closed interval and is denoted by [ a, b ]. Thus

[ a, b ] = {x : a δ x δ b}

We can also have intervals closed at one end and open at the other, i.e.,

[ a, b ) = {x : a δ x < b} is an open interval from a to b, including a but excluding b.

( a, b ] = { x : a < x δ b } is an open interval from a to b including b but excluding a.

These notations provide an alternative way of designating the subsets of set of real numbers. For example , if A = (–3, 5) and B = [–7, 9], then A ⊂ B. The set [0, ∞) defines the set of non-negative real numbers, while set ( –∞, 0 ) defines the set of negative real numbers. The set ( –∞, ∞) describes the set of real numbers in relation to a line extending from –∞ to ∞.

On real number line, various types of intervals described above as subsets of **R**, are shown in the Fig 1.1.

Here, we note that an interval contains infinitely many points.

For example, the set {x : x ∈ **R**, –5 < x ≤7}, written in set-builder form, can be written in the form of interval as (–5, 7] and the interval [–3, 5) can be written in setbuilder form as {x : –3 ≤ x < 5}. The number (b – a) is called the length of any of the intervals (a, b), [a, b], [a, b) or (a, b].