We write sets using braces and denote them with capital letters. The most natural way to describe sets is by listing all its members.

For example,

A = {1,2,3,…,10} is the set of the first 10 counting numbers, or naturals, B = {Red, Blue, Green} is the set of primary colors, N = {1,2,3,…} is the set of all naturals, and Z = {...,−3,−2,−1,0,1,2,3,…} is the set of all integers.

Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.

Some common examples of well defined sets:

- The collection of vowels in English alphabets. This set contains five elements, namely, a, e, i, o, u
- N = {1,2,3,…} is the set of counting numbers, or naturals.
- N = {1,2,3,…} is the set of counting numbers, or naturals.
- Z = {…,−3,−2,−1,0,1,2,3,…} is the set of integers.

Two sets A and B are said to be equal if and only if both the sets have same and exact number of elements. Here, if and only if means that both parts of the statement ("A = B" and "both sets have the exact same elements") are interchangeable. For example,

{2,4,6,8} = {4,8,6,2} and {2,4,6,8} = {2,4,2,6,8,2,6,4,4}.

Another example comes from the set of even naturals, which can be described as E = {2,4,6,8,…} = {2x | x ∊ N}.

A very important set is the empty set, or the null set, which has no elements. We denote the empty set by ∅, or {}. Note that we could also write, for example, ∅= {x | x ∊N and x < 0} or ∅

= {x | x ∊Q and x ∉Q}.

The intersection of sets A and B, denoted as A ∩ B, is the set of elements common to **both A AND B.**

For example:

A = {1,2,3,4,5}

B = {2,4,6,8,10}

The intersection of A and B (i.e. A∩B) is simply {2, 4}

The union of sets A and B, written as A∪B, is the set of elements that appear in **either A OR B.**

For example:

A = {1,2,3,4,5}

B = {2,4,6,8,10}

The union of A and B (i.e. A∪B) is {1, 2, 3, 4, 5, 6, 8, 10}

The difference of sets A and B, written as A-B, is the set of elements belonging to set A and **NOT** to set B.

For example:

A = {1,2,3,4,5}

B = {2,3,5}

The difference of A and B (i.e. A-B) is {1,4}

NOTE: A-B ≠ B-A

The Cartesian product of sets A and B, written A x B, is expressed as:

A x B = {(a,b)│a is every element in A, b is every element in B}

For example:

A = {1,2}

B = {4,5,6}

The Cartesian product of A and B (i.e. A x B) is {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)}

Now, lets us try doing some questions based on Set Theory.

B = {2, 4}

A = {0}

C = {1, 9, 5, 13}

D = {5, 11, 1}

E = {13, 7, 9, 11, 5, 3, 1}

F = {2, 3, 4, 5}

then n(A ∩B) = n(A) + n(B) - n(A ∪B)

= 20 + 28 - 36

= 48 - 36

= 12

(A ∪B) = 60 n(A) = 27 n(B) = 42 then;

n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

= 27 + 42 - 60

= 69 - 60 = 9

= 9

Therefore, 9 people like both tea and coffee.

B = set of persons who got medals in dramatics.

C = set of persons who got medals in music.

Given,

n(A) = 36

n(B) = 12

n(C) = 18

n(A ∪ B ∪C) = 45

n(A ∩ B ∩ C) = 4

We know that number of elements belonging to exactly two of the three sets A, B, C

= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3n(A ∩ B ∩ C)

= n(A ∩ B) + n(B ∩ C) + n(A ∩ C) - 3 × 4 ……..(i)

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩C)

Therefore, n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C)

From (i) required number

= n(A) + n(B) + n(C) + n(A ∩ B ∩ C) - n(A ∪ B ∪ C) - 12

= 36 + 12 + 18 + 4 - 45 - 12

= 70 - 67

= 3

B be the set of people who speak French.

A - B be the set of people who speak English and not French.

B - A be the set of people who speak French and not English.

A ∩ B be the set of people who speak both French and English.

Given,

n(A) = 72

n(B) = 43

n(A ∪ B) = 100

Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

= 72 + 43 - 100

= 115 - 100

= 15

Therefore, Number of persons who speak both French and English = 15

Suggested Action:

n(A) = n(A - B) + n(A ∩ B) ⇒

n(A - B) = n(A) - n(A ∩ B)

= 72 - 15

= 57

and n(B - A) = n(B) - n(A ∩ B)

= 43 - 15

= 28

Therefore, Number of people speaking English only = 57

Number of people speaking French only = 28

In this article, we have learnt about the different types of sets as well as formulas to solve questions in a simplified manner.