JEE Mathematics Sets Notes Set 01

What is a Set?

A set is a well-defined collection of objects, meaning that for any given object we can say clearly whether it belongs to the collection or not. The objects are called the elements (or members) of the set, and sets are named with capital letters A, B, C. We write x ∈ A when x is an element of A and x ∉ A when it is not.

Ways of Describing a Set

1. Roster (tabular) form: list the elements inside braces separated by commas, for example the set of primes below 12 is {2, 3, 5, 7, 11}.
2. Set-builder form: state the rule that the elements obey, for example A = {x : x is a capital city of India} or Z = {x : x is an integer}. The set {1, 4, 9, 16, 25} can be written as {x² : x ∈ N, x ≤ 5}.

Cardinal Number and Kinds of Sets

The number of elements in a set A is its cardinal number n(A). For A = {a, b, c, d} we have n(A) = 4. A set with a limited number of elements is finite, such as the months of a year. A set that is not finite is infinite, such as the set of points on a line or the set of natural numbers. The set with no element is the empty (null) set ∅, and n(∅) = 0.

Subsets

A is a subset of B, written A ⊆ B, when every element of A is also in B. If in addition B has at least one element that A lacks, A is a proper subset of B, written A ⊂ B. The empty set is a subset of every set, and every set is a subset of itself. If A ⊆ B and B ⊆ A then A = B. A set with n elements has 2ⁿ subsets, of which 2ⁿ − 1 are proper. For the usual number sets N ⊂ Z ⊂ Q ⊂ R.

Operations on Sets

1. Intersection A ∩ B = {x : x ∈ A and x ∈ B}, the elements common to both.
2. Union A ∪ B = {x : x ∈ A or x ∈ B or both}.
3. Difference A − B = {x : x ∈ A and x ∉ B}.
4. Complement A′ = {x ∈ U : x ∉ A}, taken inside a universal set U.
5. Two sets are disjoint when they have no common element, that is A ∩ B = ∅.

Counting Formulas

For two finite sets n(A ∪ B) = n(A) + n(B) − n(A ∩ B). When A and B are disjoint the last term is zero. De Morgan's laws state (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Worked Examples

Example 1. A is the set of even numbers greater than 5 and less than 13, B is the set of divisors of 70, and C is the set of primes less than 17. Write them in roster form and find A ∪ B, B ∩ C, (A ∪ B) ∩ C and A ∩ (B ∩ C).
A = {6, 8, 10, 12}, B = {1, 2, 5, 7, 10, 14, 35, 70}, C = {2, 3, 5, 7, 11, 13}.
A ∪ B = {1, 2, 5, 6, 7, 8, 10, 12, 14, 35, 70}; B ∩ C = {2, 5, 7}; (A ∪ B) ∩ C = {2, 5, 7}; A ∩ (B ∩ C) = ∅, since none of 6, 8, 10, 12 lies in {2, 5, 7}.

Example 2. Decide whether 13 × 0 ∈ Z and \( \sqrt{2} \cdot \sqrt{8} \in Z \).
13 × 0 = 0, which is an integer, so the first statement is true. \( \sqrt{2} \cdot \sqrt{8} = \sqrt{16} = 4 \), an integer, so the second is also true.

Example 3. In a group of 40 people, 25 like tea, 20 like coffee and 5 like neither. How many like both?
n(T ∪ C) = 40 − 5 = 35. From the counting formula 35 = 25 + 20 − n(T ∩ C), so n(T ∩ C) = 10.