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 supersetof 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 ∈ Rand 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].
