Subsets

Introduction

Subsets define the concepts of relations and functions. Knowledge of subsets is required in geometry, sequences, probability, etc. A set is a well-defined collection of objects represented as {A, B, C, D, X, Y, Z}. The elements of sets are separated by commas and enclosed within brackets { }.

If X is the set of all triangles and Y is the set of all equilateral triangles, it means every element of Y is an element of X. We can conclude that Y is a subset of X.

Subset meaning

Set ‘A’ is a subset of set ‘B’ if every element of ‘A’ is also an element of ‘B.’

If set A is a subset of set B, it is represented as A ⊂ B, where ‘⊂’ means ‘is a subset of.’

If a ∈ A and a ∈ B, ∈ (epsilon) denotes the phrase ‘belongs to.’

A ⊂ B if a ∈ A ⇒ a ∈ B

This means that A is a subset of B. If a is an element of A, it implies that a is also an element of B.

If A is a subset of B, B is a superset of A.

If A is not a subset of B, it is represented as A ⊄ B.

Subset examples

1. If A = set of natural numbers = {1, 2, 3, 4,…}

B = set of whole numbers = {0, 1, 2, 3, 4,..}

Every element of A is an element of B, so A is a subset of B. This means A ⊂ B.

2. If X = {p, q, r} and Y = set of all alphabets. Then, X is a subset of Y. This is represented as X ⊂ Y.
3. If Q = set of rational numbers and R = set of real numbers. Then, Q is a subset of the set R. This means Q ⊂ R.
4. If A = {1, 3, 5} and B {x : x is an odd natural number less than 6}. Then, A ⊂ B and B ⊂ A, hence A=B.
5. If A = vowels = {a, e, i, o, u}

   B = Consonant {b,c,d,f…}

   A is not a subset of B. Also, B is not a subset of A.

6.Consider, A = {a, b} B = {c, d} C = {a, b, c, d}

A ⊂ C, B ⊂ C, φ ⊂ A, φ ⊂ B, φ ⊂ C

But A is not a subset of B, A ⊄ B.

Types of subsets

There are various types of subsets:

1. Proper subset – Any subset of the set except itself. For example, A = {1, 2, 3}, then its proper subsets can be {1, 2}, {2,3}, {1,3}. {3, 1} but the set {1, 2, 3} itself is not a proper subset of A. If A is a proper subset of B, then A ⊂ B and A ≠ B. It contains only a few elements of set A, so it is never equal to set A.
2. Improper subset – Every set has one improper subset, the set itself. For the subset {1, 2, 3}, it’s only improper subset is {1, 2, 3}. {a, b} is the only improper subset of {a, b}. If A is an improper subset of B, then A ⊆ B and contains all elements of set A, hence always equal to set A.
3. Singleton subset – If a set A has a single element, it is termed a singleton subset. For example, if set A has one element {a}, then {a} is a singleton subset.

The subset of a set of real numbers

The set of natural numbers = N = {1, 2, 3, 4, 5, 6, 7,……}

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}

The set of irrational numbers = T = {x : x ∈ R and x ∉ Q}

Q is the set of all numbers x, p and q are the integers and q is not equal to zero. T is composed of all real numbers that are not rational numbers like √11, √7 and π.

The relations among the subset of a set of real numbers:

N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T

Intervals as subsets of R

a, b ∈ R and a < b

Open intervals are denoted by (a, b). It is represented by { y : a < y < b}. All the points between a and b belong to open intervals but a and b themselves don’t belong to open intervals. Example: A = { x : 9 < x < 10}. A = (9, 10)

Closed intervals are denoted by [a, b]. It contains the endpoints and is represented as {x : a ≤ x ≤ b}. Example: A = { x : 9 ≤ x ≤ 10}. A = [9, 10]

If a subset contains open interval and closed intervals, then it is represented as:

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

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

Example: A = { x : 4 ≤ x < 5}. A = [4, 5)

         A = { x : 4 < x ≤ 5}. A = (4, 5]

On the natural number line, the subset of R is represented as:

(b – a) is the length of any of the intervals (a, b), [a, b], [a, b) or (a,b].

Subset formulas

2n = number of subsets

2n – 1 = number of proper subsets

1 = number of improper subsets

Example: If A = {2, 3, 5}, then how many proper subsets does X have?

Solution: A contains three elements, so n = 3

         The number of proper subsets of A are = 23 – 1 = 8 – 1 = 7

         Proper subset A can be any subset except itself. So proper subsets are:

         {}, {2}, {3}, {5}, {2, 3}, {2, 5}, {3,5}

Answer: A has 7 proper subsets: {}, {2}, {3}, {5}, {2, 3}, {2, 5}, {3,5}

Conclusion

A set is a well-defined collection of objects. The number of the sets is 2n (n=number of elements in the set). A subset is a part of a given set that can be the same or another set. A proper subset is a set with different combinations of elements except for the set. Subsets and supersets are related to each other. If A is a subset of B, B is A’s superset.