# Set Theory

A set is a well-defined collection of objects or elements. Each element in a set is unique. Usually but not necessarily a set is denoted by a capital letter e.g., A, B, ....., U, V etc. and the elements are enclosed between brackets { }, denoted by small letters a, b, ....., x, y etc.

A = Set of all small English alphabets

= {a, b, c, ....., x, y, z}

B = Set of all positive integers less than or equal to 10

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

R = Set of real numbers

= {x : - âˆž < x < âˆž}

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The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers). The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by Î¦(phi). The number of elements of a set A is denoted as n(A) and hence n(Î¦) = 0 as it contains no element.

# Union of Sets

Union of two or more sets is the best of all elements that belong to any of these sets. The symbol used for union of sets is 'U'

i.e. AUB = Union of set A and set B

= {x : x Îµ A or x Îµ B (or both)}

**e.g.** If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8} then AUBUC = {1, 2, 3, 4, 5, 6 8}.

# Intersection of Sets

It is the set of all of the elements, which are common to all the sets. The symbol used for intersection of sets is 'âˆ©'.

i.e. Aâˆ©B = {x : x Îµ A and x Îµ B}

**Example:** If A = {1, 2, 3, 4} an B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then Aâˆ©Bâˆ©C = {2}.

Remember that n(AUB) = n(A) + n(B) -(Aâˆ©B).

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