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**Relation**

A relation, R, from a non-empty set A to another non-empty set B, is a subset of A Ã— B.

The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A Ã— B.

**Ex**:** **A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}. Find a relation R from A to B, such that for any *a* âˆˆ A and *b* âˆˆ B, '*a*' is factor of '*b*' and *a* < *b*.

**Solution**: Since A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54},

we have to find the set of ordered pairs (*a*, *b*) such that *a* is factor of *b* and *a* < *b*.

Let 2 âˆˆ A, 4 âˆˆ B, as 2 is a factor of 4 and 2 < 4.

So (2, 4) is one such ordered pair.

Likewise, (2, 6), (2, 18), (2, 54)â€¦ are other such ordered pairs.

Thus, the required relation is R = {(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}.

Domain of R = {2, 6, 9}

Range of R = {4, 6, 18, 27, 54}

If (*a*, *b*) âˆˆ R, we say that *a* is related to *b*, and we denote it as *a* R *b*.

If n (A) = p, n (B) = q, then n (A Ã— B) = pq, and the total number of possible relations from set A to set B is 2^{pq } .

**Empty Relation**

A relation R in a set A is called an empty relation, if no element of A is related to any element of A, i.e. R = âˆ… âˆˆ A Ã— A.

Let R be a relation defined as R = {(x, y)/x+y â‰¤ 8} in A

Choose any arbitrary ordered pair from A cross A. Here, all the ordered pairs satisfy the condition that the sum of their elements is less than or equal to 8.

Hence, R = A Ã— A.

**Universal Relation**

A relation R in a set A is called a universal relation or total relation, if every element of A is related to every element of A, i.e. R = A Ã— A.

An empty relation and the universal relation are called trivial relations.

Different types of relations are:

i) Reflexive

ii) Symmetric

iii) Transitive

iv) Equivalence

**Reflexive Relation**

Let T be the set of all triangles in a plane.

R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

If (T_{1}, T_{2}) âˆˆ R, â‡’ T_{1 }is similar to T_{1}, which is true.

â‡’ Every element of set R is related to itself.

R in T is reflexive.

A relation, R in T, is said to be reflexive if every element of set R is related to itself, i.e. (*x, x*) âˆˆ R, for every x âˆˆ T.

**Symmetric Relation**

Let T be the set of all triangles in a plane.

R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

Let (T_{1}, T_{2}) âˆˆ R, â‡’ T_{1 }is similar to T_{2}

â‡’ T_{2} is similar to T_{1}

â‡’ (T_{1}, T_{2}) âˆˆ R

A relation R in a set T is said to be symmetric if (x,_{ }y) âˆˆ R implies that (y, x) âˆˆ R âˆ€ x, y âˆˆ T.

Example: Relation R = {(1, 2), (2, 1), (1, 3)} in the set, {1, 2, 3}, is not symmetric.

'.' (1, 3) âˆˆ R but (3, 1) âˆ‰ R

Note: It is wrong to say that it is "sometimes symmetric" or "is symmetric as far as 1 and 2 are concerned", since being symmetric is a property of the whole relation.

**Transitive Relation**

A Relation, R, in T is said to be transitive, if (x, y) âˆˆ R and (y, z) âˆˆ R imply that (x, z) âˆˆ R âˆ€ x, y, z âˆˆ T.

Let T be the set of all triangles in a plane. R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

Let T_{1}, T_{2} and T_{3 }âˆˆ T such that (T_{1} ,T_{2}) âˆˆ R and (T_{2} ,T_{3}) âˆˆ R.

â‡’ T_{1} is similar to T_{2} and T_{2} is similar to T_{3}

â‡’ T_{1} is similar to T_{3}

â‡’ (T_{1} ,T_{3}) âˆˆ R

Hence, (T_{1} ,T_{2}) âˆˆ R and (T_{2} ,T_{3}) âˆˆ R â‡’ (T_{1} ,T_{3}) âˆˆ R

âˆ´ Relation, R, in T is transitive.

Let T be the set of all triangles in a plane. R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

The relation, R in T, is reflexive, symmetric and transitive.

Relation R in T is an equivalence relation.

** Equivalence relation:** A relation is said to be equivalence relation if it is reflexive, symmetric and transitive.

**Relation**

A relation, R, from a non-empty set A to another non-empty set B, is a subset of A Ã— B.

The subset, R, is derived by describing a relationship between the first element and the second element of the ordered pairs in A Ã— B.

**Ex**:** **A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}. Find a relation R from A to B, such that for any *a* âˆˆ A and *b* âˆˆ B, '*a*' is factor of '*b*' and *a* < *b*.

**Solution**: Since A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54},

we have to find the set of ordered pairs (*a*, *b*) such that *a* is factor of *b* and *a* < *b*.

Let 2 âˆˆ A, 4 âˆˆ B, as 2 is a factor of 4 and 2 < 4.

So (2, 4) is one such ordered pair.

Likewise, (2, 6), (2, 18), (2, 54)â€¦ are other such ordered pairs.

Thus, the required relation is R = {(2, 4), (2, 6), (2, 18), (2, 54), (6, 18), (6, 54,), (9, 18), (9, 27), (9, 54)}.

Domain of R = {2, 6, 9}

Range of R = {4, 6, 18, 27, 54}

If (*a*, *b*) âˆˆ R, we say that *a* is related to *b*, and we denote it as *a* R *b*.

If n (A) = p, n (B) = q, then n (A Ã— B) = pq, and the total number of possible relations from set A to set B is 2^{pq } .

**Empty Relation**

A relation R in a set A is called an empty relation, if no element of A is related to any element of A, i.e. R = âˆ… âˆˆ A Ã— A.

Let R be a relation defined as R = {(x, y)/x+y â‰¤ 8} in A

Choose any arbitrary ordered pair from A cross A. Here, all the ordered pairs satisfy the condition that the sum of their elements is less than or equal to 8.

Hence, R = A Ã— A.

**Universal Relation**

A relation R in a set A is called a universal relation or total relation, if every element of A is related to every element of A, i.e. R = A Ã— A.

An empty relation and the universal relation are called trivial relations.

Different types of relations are:

i) Reflexive

ii) Symmetric

iii) Transitive

iv) Equivalence

**Reflexive Relation**

Let T be the set of all triangles in a plane.

R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

If (T_{1}, T_{2}) âˆˆ R, â‡’ T_{1 }is similar to T_{1}, which is true.

â‡’ Every element of set R is related to itself.

R in T is reflexive.

A relation, R in T, is said to be reflexive if every element of set R is related to itself, i.e. (*x, x*) âˆˆ R, for every x âˆˆ T.

**Symmetric Relation**

Let T be the set of all triangles in a plane.

R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

Let (T_{1}, T_{2}) âˆˆ R, â‡’ T_{1 }is similar to T_{2}

â‡’ T_{2} is similar to T_{1}

â‡’ (T_{1}, T_{2}) âˆˆ R

A relation R in a set T is said to be symmetric if (x,_{ }y) âˆˆ R implies that (y, x) âˆˆ R âˆ€ x, y âˆˆ T.

Example: Relation R = {(1, 2), (2, 1), (1, 3)} in the set, {1, 2, 3}, is not symmetric.

'.' (1, 3) âˆˆ R but (3, 1) âˆ‰ R

Note: It is wrong to say that it is "sometimes symmetric" or "is symmetric as far as 1 and 2 are concerned", since being symmetric is a property of the whole relation.

**Transitive Relation**

A Relation, R, in T is said to be transitive, if (x, y) âˆˆ R and (y, z) âˆˆ R imply that (x, z) âˆˆ R âˆ€ x, y, z âˆˆ T.

Let T be the set of all triangles in a plane. R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

Let T_{1}, T_{2} and T_{3 }âˆˆ T such that (T_{1} ,T_{2}) âˆˆ R and (T_{2} ,T_{3}) âˆˆ R.

â‡’ T_{1} is similar to T_{2} and T_{2} is similar to T_{3}

â‡’ T_{1} is similar to T_{3}

â‡’ (T_{1} ,T_{3}) âˆˆ R

Hence, (T_{1} ,T_{2}) âˆˆ R and (T_{2} ,T_{3}) âˆˆ R â‡’ (T_{1} ,T_{3}) âˆˆ R

âˆ´ Relation, R, in T is transitive.

Let T be the set of all triangles in a plane. R is a relation in T defined by

R = {(T_{1}, T_{2}): T_{1 }is similar to T_{2}}

The relation, R in T, is reflexive, symmetric and transitive.

Relation R in T is an equivalence relation.

** Equivalence relation:** A relation is said to be equivalence relation if it is reflexive, symmetric and transitive.