Relations

We encounter different types of relationship in our daily life. Besides human relationship that we are so familiar with, there are numerous other relationships, including those which are purely abstraction of processes and ideas.

In essence, any two elements which are paired have potential to possess relationship between them. Now think of the Cartesian product that we have defined for two sets. It consists of ordered pairs of elements - one each from the two sets. The total numbers of ordered pairs in a Cartesian product is equal to the product of numbers of elements in each set. A particular relation, however, may not comprise all ordered pairs.

In this module, we shall limit our discussion to binary relations only. A binary relation is a relation as defined between two elements either from the same set of from two different sets.

Consider a “get together”. Divide the people in two groups comprising of males and females. A certain numbers of ordered pairs will qualify for a relation say “classmate of” – not all. Similarly, a relation such as “brother of” may include some of the ordered pairs of the Cartesian product of two set.

$$M=\{A,B,C,D,E\}$$

$$F=\{G,H,I,J,K\}$$

We can represent the relationship of “classmate of” as shown here :

From the figure, we can write the collection of relationship “classmate of” as a set of ordered pairs of two sets,

$$R\left(\text{classmate of}\right)=\{\left(A,G\right),\left(A,K\right),\left(D,G\right),\left(D,I\right)\}$$

In the nutshell, we can think of relation as a collection (set), which comprises of ordered pairs (instances of relation). Note that it is a specific relation. This is a relation between elements of two sets. Clearly, this relation set can not exceed the Cartesian product of two sets under consideration. Thus, a relation set is a subset of Cartesian product set.

Relation

A relation “R” from a non-empty set “A” to non-empty set “B” is a subset of Cartesian product “AXB”.

We need to note that a relation is defined in a particular order “from” to “to”. It is for this reason, we denoted relation pictorially by an arrow which is directed from the elements of "from" set “A” to "to" set “B”.

Consider few examples,

$$R=\{\left(x,y\right):x=y^2,x\in A,y\in B\}$$

$$R=\{\left(x,y\right):x=y+\mathrm{1,}x\in A,y\in B\}$$

In certain circumstance, we are required to work with relation among the elements of the same set. For example, consider the male set defined earlier. Some of the male members may as well be classmates and hence related to each other. Such relation is relation on one set only and is called "relation on A" or "relation on B" etc. Few examples are :

$$R=\{\left(x,y\right):x=x+\mathrm{1,}x\in A\}$$

$$R=\{\left(x,y\right):y=x+1andx,y\in A\}$$

Example

Problem 1 : Let $A=\{\mathrm{1,2,}\dots ..,10\}$ . Write down the relation set in roaster form, which is defined as :

$$R=\{\left(x,y\right):y=3xandx,y\in A\}$$

Solution : We begin with the first element of “A” i.e. x =1. Since other element also belongs to set “A”, it is required that the value of “y” be one of the elements in the set "A".

$$Forx=\mathrm{1,}y=3X1=3$$

$$Forx=\mathrm{2,}y=3x=3X2=6$$

$$Forx=\mathrm{3,}y=3x=3X3=9$$

Thus, “x” can assume values “1”,”2” and “3” for which “y” can assume values “3”,”6” and “9” respectively in accordance with the given relation. The relation, therefore, is set of ordered pairs :