- What is Cartesian Sets
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- What is relations?
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- What is Function
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- Algebra of Real Function
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- Type Of Relations
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- Type Of Functions
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- Composite Function
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- Invertible Function

Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R = {(a, b) | a divides b}

We saw that R was reflexive since every number divides itself

Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R ={(1,1),(2,2),(2,3),(3,2),(4,4)}

We saw that R is not reflexive since every number is not present in R

Let A = {1, 2, 3, 4} and R = {(1, 2), (2,1),(3,3),(4,4)}

This relation is symmetric. It satisfies the above criterion.

Let A = Z and define R = {(a, b) | a > b}.

R is transitive because if a > b and b > c then a > c.

Note that transitivity, like symmetry, is possessed by a relation unless the stated condition is violated. So unless you can find pairs (a, b) and (b, c) which are in R while (a, c) is not, then the relation is transitive. In particular, the empty relation is always transitive because it has no pairs to violate the condition

Example:

Let A = Z and define R = {(a, b) | 3 divides a - b}.

It is reflexive as (a-a) will always be divided by 3

It is symmetric as if (a-b) will be divided by 3 ,the (b-a) will also divided by 3

it is transitive as if (a,b) ∈ R and (b,) ∈ R which means a-b and b-c are divided by 3, now a-c=(a-b)+(b-c) ,so (a,c) ∈ R

Determine if the below relation is an equivalence Relation

A={1,2,3,4,5,.....,14)

R is the relation on set A defined as

R={(a,b)| (3a-b)=0}

From the defination , R would contain following elements

R={(1,3),(2,6),(3,9),(4,12)}

R is not reflexive as we dont have (1,1),(2,2) like that in the Relation

R is not symmetric also as we dont (3,1) for (1,3) in the relation

R is not transitive also as dont have (1,9) for (1,3), (3,9) in R

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