Author: admin
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Equivalence Relations
A relation R on a set A is called an equivalence relation if it satisfies following three properties: Example: Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}. Show that R is an Equivalence Relation. Solution: Reflexive: Relation R is…
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Closure Properties of Relations
Consider a given set A, and the collection of all relations on A. Let P be a property of such relations, such as being symmetric or being transitive. A relation with property P will be called a P-relation. The P-closure of an arbitrary relation R on A, indicated P (R), is a P-relation such that…
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Types of Relations
1. Reflexive Relation: A relation R on set A is said to be a reflexive if (a, a) ∈ R for every a ∈ A. Example: If A = {1, 2, 3, 4} then R = {(1, 1) (2, 2), (1, 3), (2, 4), (3, 3), (3, 4), (4, 4)}. Is a relation reflexive? Solution: The relation is reflexive as for every…
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Composition of Relations
Let A, B, and C be sets, and let R be a relation from A to B and let S be a relation from B to C. That is, R is a subset of A × B and S is a subset of B × C. Then R and S give rise to a relation…
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Representation of Relations
Relations can be represented in many ways. Some of which are as follows: 1. Relation as a Matrix: Let P = [a1,a2,a3,…….am] and Q = [b1,b2,b3……bn] are finite sets, containing m and n number of elements respectively. R is a relation from P to Q. The relation R can be represented by m x n matrix…
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Binary Relation
Let P and Q be two non- empty sets. A binary relation R is defined to be a subset of P x Q from a set P to Q. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e., aRb. If sets P and Q…
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Inclusion-Exclusion Principle
Let A, B be any two finite sets. Then n (A ∪ B) = n (A) + n (B) – n (A ∩ B) Here “include” n (A) and n (B) and we “exclude” n (A ∩ B) Example 1: Suppose A, B, C are finite sets. Then A ∪ B ∪ C is finite…
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Multisets
A multiset is an unordered collection of elements, in which the multiplicity of an element may be one or more than one or zero. The multiplicity of an element is the number of times the element repeated in the multiset. In other words, we can say that an element can appear any number of times…
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Algebra of Sets
Sets under the operations of union, intersection, and complement satisfy various laws (identities) which are listed in Table 1. Table: Law of Algebra of Sets Idempotent Laws (a) A ∪ A = A (b) A ∩ A = A Associative Laws (a) (A ∪ B) ∪ C = A ∪ (B ∪ C) (b) (A…
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Operations on Sets
The basic set operations are: 1. Union of Sets: Union of Sets A and B is defined to be the set of all those elements which belong to A or B or both and is denoted by A∪B. Example: Let A = {1, 2, 3}, B= {3, 4, 5, 6}A∪B = {1, 2, 3,…