Set Theory

i. Set Theory: A Set is defined as a group of objects, known as elements. These objects could be anything conceivable, including numbers, letters, colors, even set themselves. However, none of the objects of the set can be the set itself.

ii. Set Notation: 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.

iii. Well-defined Set: 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.

iv. Set Equality: 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 â?? N}.

v. Null Set: 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 â??N and x < 0} or ∅ = {x | x â??Q and x ∉Q}.

vi. Intersection of Sets: 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}

vii. Union of Sets: 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}

viii. Difference of Sets: 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

ix. Cartesian Product of Sets: The Cartesian product of sets A and B, written A x B, is expressed as: A x B = {(a,b)â??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)}.