Set Operations

MathematicsComputer EngineeringMCA

Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets.

Examples

Set Operations

Set Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.

Set Union

The union of sets A and B (denoted by A ∪ B) is the set of elements that are in A, in B, or in both A and B. Hence, A ∪ B = { x | x ∈ A OR x ∈ B }.

Example − If A = { 10, 11, 12, 13 } and B = { 13, 14, 15 }, then A ∪ B = { 10, 11, 12, 13, 14, 15 }. (The common element occurs only once)

Set Intersection

The intersection of sets A and B (denoted by A ∩ B) is the set of elements which are in both A and B. Hence, A ∩ B = { x | x ∈ A AND x ∈ B }.

Example − If A = { 11, 12, 13 } and B = { 13, 14, 15 }, then A ∩ B = { 13 }.

Set Difference/ Relative Complement

The set difference of sets A and B (denoted by A – B) is the set of elements that are only in A but not in B. Hence, A - B = { x | x ∈ A AND x ∉ B }.

Example − If A = { 10, 11, 12, 13 } and B = { 13, 14, 15 }, then (A - B) = { 10, 11, 12 } and (B - A) = { 14, 15 }. Here, we can see (A - B) ≠ (B - A)

Complement of a Set

The complement of a set A (denoted by A’) is the set of elements which are not in set A. Hence, A' = { x | x ∉ A }.

More specifically, A'= (U - A) where U is a universal set that contains all objects.

Example − If A = { x | x belongs to set of odd integers } then A' = { y | y does not belong to set of odd integers }

Cartesian Product / Cross Product

The Cartesian product of n number of sets A1, A2, ... An denoted as A1 × A2 ... × An can be defined as all possible ordered pairs (x1, x2, ... xn) where x1 ∈ A1, x2 ∈ A2, ... xn ∈ A_n

Example − If we take two sets A = { a, b } and B = { 1, 2 },

The Cartesian product of A and B is written as − A × B = { (a, 1), (a, 2), (b, 1), (b, 2)}

The Cartesian product of B and A is written as − B × A = { (1, a), (1, b), (2, a), (2, b)}

raja
Published on 26-Aug-2019 10:36:42
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