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Relations - Lesson Summary

A relation gives a connection or an association between objects, ideas, quantities or individuals.

Relations also exist between groups of objects, ideas, quantities and individuals, which form sets.

Let S = {S1, S2, S3} and D = { D1, D2}

The Cartesian product of S and D can be obtained by pairing each element of set S with the elements of set D in a specific order.

S × D = {(S1, D1), (S1, D2), (S2, D1), (S2, D2), (S3, D1), (S3, D2)}

Let X = {Mathematics, Biology, Chemistry}, Y = {Sets, Atoms, Cells, Circles, Force}

The possible combinations of the subjects and the topics are the set of ordered pairs that can be obtained by the Cartesian product of sets X and Y.

The total number of such possible ordered pairs is 15.

X x Y = {(Mathematics,Sets),(Mathematics,Atoms),(Mathematics,Cells),(Mathematics,Circles),(Mathematics,Force),(Biology,Sets),(Biology,Atoms),(Biology,Cells),(Biology,circles),(Biology,Force),(Chemistry,Sets),(Chemistry,Atoms),(Chemistry,Cells),(Chemistry,circles),(Chemistry,Force)}

A relation can be established between the first and the second elements of the ordered pair such that the second element is a topic of the first element.

R = {(x,y): y is a topic of x,x ∈ X,y ∈ Y}

All ordered pairs x-y, such that y is a topic of x, where x belongs to set X and y belongs to set Y. This is the set-builder representation of the relation R.

In the roster form, we list all the ordered pairs that satisfy the formula given in the relation. It has four ordered pairs; {(Mathematics,Sets),(Mathematics,Circles),(Biology,Cells),(Chemistry,Atoms)} .

A relation can be described in different ways:
1) List form or Roster form
2) Set-builder form
3) Arrow diagram (visual representation)

Definition of a relation
A relation from a non-empty set A to a non-empty set B can be defined as the subset of A x B, which is obtained by describing a relationship between the first element and the second element of the ordered pairs.

The property that explains how the elements in each ordered pair are related is called the rule of relation or the formula of relation.

R = {(x,y):y is a topic of x,x ∈ X,y ∈ Y} = {(Mathematics,Sets),(Mathematics,Circles),(Biology,Cells),(Chemistry,Atoms)} .

The set of the first coordinates of all the ordered pairs of a relation R is called the domain of R.

Domain = {Mathematics, Biology, Chemistry}

The set of the second coordinates of all the ordered pairs of R is called the range of R.

Range = {Sets, Atoms, Cells, Circles}

Codomain is Y= {Sets, Atoms, Cells, Circles, Force}

Range ⊆ Codomain

The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A × B.

If n (A) = p and n (B) = q, then n (A × B) = pq and the total number of relations is 2pq.

Possible number of relations that can be defined from set X to Y:

X = {Mathematics, Biology, Chemistry}, Y = {Sets, Atoms, Cells, Circles, Force}

n(X) = 3 and n(Y) = 5, n(A x B) = 15

∴  Total number of relations = 215


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