Types of relations

1. TYPES OF A RELATION

2. Types of a relation
• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation
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3. Types of a relation
• Let us consider A = {2, 3, 4}
• Then the Cartesian product
A X A = {(2,2),(2,3),(2,4),(3,2),(3,3),(3,4),(4,2),(4,3),(4,4)}
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
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4. (A) REFLEXIVE RELATION
• A relation R is called reflexive if every element of the relation is
related to itself.
• It is denoted by xRx
• For example: R1= {(2, 2), (3 ,3),(4, 4)
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
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5. (B) SYMMETIC RELATION
• In a relation, if the first and second components of the ordered pairs are
interchanged, the relation still holds. It is called symmetric relation.
• It is written as is xRy then yRx.
• If x = y then the relation is Anti-symmetric.
• For example R2= {(2, 3), (3, 2)
R3 = {(2,3),(3,2),(2,4),(4,2)}
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
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6. ( C) TRANSITIVE RELATION
• A relation is called transitive if aRb and bRc gives aRc. Then a is
related to b and b is related to c the a is related to c.
• For example: {(2, 3), (3, 4), (2, 4)}
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
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7. (D) EQUIVALENCE RELATION
• A relation is called equivalence relation if and only if it is reflexive,
symmetric and transitive.
• For example: R4 = {(2, 2),(2,3),(2,4),(3,2),(3,3),(3,4),(4,4)}
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
A
A
X 2 3 4
2 (2,2) (2, 3) (2, 4)
3 (3,2) (3,3) (3, 4)
4 (4,2) (4, 3) (4 , 4)
Reflexive relation Symmetric relation Transitive relation
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8. Inverse Relation
Here, A×B = {(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3,2), (3,3),(3,4)}
A relation “is less than” from set A to set B in the arrow diagram is
1
2
3
A
2
3
4
B
R
2
3
4
B
1
2
3
A
Inverse Relation
R-1
R = {(1, 2), (1,3), (1,4), (2,3), (2,4), (3,4)} R-1 = {(2,1),(3, 1), (4,1), (3,2), (4,2), (4, 3)}

9. Inverse Relation
1
2
3
A
2
3
4
B
R
2
3
4
B
1
2
3
A
Inverse Relation
R-1
R = {(1, 2), (1,3), (1,4), (2,3), (2,4), (3,4)} R-1 = {(2,1),(3, 1), (4,1), (3,2), (4,2), (4, 3)}
R = “is less than” relation R-1 = “is greater than” relation
Symbolically: If R = {(x, y): x∈A, y ∈B}, then R-1 = {(y, x): y ∈ B and x∈A}

10. A relation is defined by R = {(x, y): 1< x < 4; y = 2x – 1}. Find the domain and range
of the function . Also find the inverse relation
• Domain x ∈ {2, 3}
• Range: when x = 2, y = 2x 2 – 1 = 3
• when x = 3, y = 2 x 3 – 1 = 5
• ∴Range = {3, 5]
• So, relation R = {(2, 3), (3, 5)}
Hence, Inverse relation R-1{(3, 2), (5, 3)}

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