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## A Trip to Mathematics: Part III Relations and Functions

Read these statements carefully:
‘Michelle is the wife of Barak Obama.’
‘John is the brother of Nick.’
‘Robert is the father of Marry.’
‘Ram is older than Laxman.’
‘Mac is the product of Apple Inc.’
After reading these statements, you will realize that first ‘Noun’ of each sentence is some how related to other. We say that each one noun is in a RELATIONSHIP to other. Mischell is related to Barak Obama, as wife. John is related to Nick, as brother. Robert is related to Marry, as father. Ram is related to Laxman in terms of age(seniority). Mac is related to Apple Inc. as a product.These relations are also used in Mathematics, but a little variations; like ‘alphabets’ or ‘numbers’ are used at place of some noun and mathematical relations are used between them. Some good examples of relations are:

is less than
is greater than
is equal to
is an element of
belongs to
divides
etc. etc.

Some examples of regular mathematical statements which we encounter daily are:

4<6 : 4 is less than 6.
5=5 : 5 is equal to 5.
6/3 : 3 divides 6.

For a general use, we can represent a statement as:
”some x is related to y”
Here ‘is related to’ phrase is nothing but a particular mathematical relation. For mathematical convenience, we write ”x is related to y” as $x \rho y$. x and y are two objects in a certain order and they can also be used as ordered pairs (x,y).
$(x,y) \in \rho$ and $x \rho y$ are the same and will be treated as the same term in further readings. If $\rho$ represents the relation motherhood, then $\mathrm {(Jane, \ John)} \in \rho$ means that Jane is mother of John.
All the relations we discussed above, were in between two objects (x,y), thus they are called Binary Relations. $(x,y) \in \rho \Rightarrow \rho$ is a binary relation between a and b. Similarly, $(x,y,z) \in \rho \Rightarrow \rho$ is a ternary (3-nary) relation on ordered pair (x,y,z). In general a relation working on an n-tuple $(x_1, x_2, \ldots x_n) \in \rho \Rightarrow \rho$ is an n-ary relation working on n-tuple.
We shall now discuss Binary Relations more rigorously, since they have solid importance in process of defining functions and also in higher studies. In a binary relation, $(x,y) \in \rho$, the first object of the ordered pair is called the the domain of relation ρ and is defined by
$D_{\rho} := \{x| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$ and also the second object is called the range of the relation ρ and is defined by $R_{\rho} := \{y| \mathrm{for \ some \ y, \ (x,y) \in \rho} \}$.
There is one more thing to discuss about relations and that is about equivalence relation.
A relation is equivalence if it satisfies three properties, Symmetric, Reflexive and Transitive.
I mean to say that if a relation is symmetric, reflexive and transitive then the relation is equivalence. You might be thinking that what these terms (symmetric, reflexive and transitive) mean here. Let me explain them separately:
A relation is symmetric: Consider three sentences “Jen is the mother of John.”; “John is brother of Nick.” and “Jen, John and Nick live in a room altogether.”
In first sentence Jen has a relationship of motherhood to John. But can John have the same relation to Jen? Can John be mother of Jen? The answer is obviously NO! This type of relations are not symmetric. Now consider second statement. John has a brotherhood relationship with Nick. But can Nick have the same relation to John? Can Nick be brother of John? The answer is simply YES! Thus, both the sentences “John is the brother of Nick.” and “Nick is the brother of John.” are the same. We may say that both are symmetric sentences. And here the relation of ‘brotherhood’ is symmetric in nature. Again LIVING WITH is also symmetric (it’s your take to understand how?).
Now let we try to write above short discussion in general and mathematical forms. Let X and Y be two objects (numbers or people or any living or non-living thing) and have a relation ρ between them. Then we write that X is related by a relation ρ , to Y. Or X ρ Y.
And if ρ is a symmetric relation, we might say that Y is (also) related by a relation ρ to X. Or Y ρ X.
So, in one line; $X \rho Y \iff Y \rho X$ is true.

A relation is reflexive if X is related to itself by a relation. i.e., $X \rho X$. Consider the statement “Jen, John and Nick live in a house altogether.” once again. Is the relation of living reflexive? How to check? Ask like, Jen lives with Jen, true? Yes! Jen lives there.
A relation is transitive, means that some objects X, Y, Z are such that if X is related to Y by the relation, Y is related to Z by the relation, then X is also related to Z by the same relation.
i.e., $X \rho Y \wedge Y \rho Z \Rightarrow X \rho Z$. For example, the relationship of brotherhood is transitive. (Why?) Now we are able to define the equivalence relation.
We say that a relation ρ is an equivalence relation if following properties are satisfied: (i) $X \rho Y \iff Y \rho X$
(ii) $X \rho X$
(iii) $X \rho Y \ Y \rho Z \Rightarrow X \rho Z$.

Functions: Let f be a relation (we are using f at the place of earlier used ρ ) on an ordered pair $(x,y) : x \in X \ y \in Y$. We can write xfy, a relation. This relation is called a function if and only if for every x, there is always a single value of y. I mean to say that if $xfy_1$ is true and $xfy_2$ is also true, then always $y_1=y_2$. This definition is standard but there are some drawbacks of this definition, which we shall discuss in the beginning of Real Analysis .
Many synonyms for the word ‘function’ are used at various stages of mathematics, e.g. Transformation, Map or Mapping, Operator, Correspondence. As already said, in ordered pair (x,y), x is called the element of domain of the function (and X the domain of the function) and y is called the element in range or co-domain of the function (and Y the range of the function).

Here I will stop myself. I don’t want a post to be long (specially when writing on basic mathematics) that reader feel it boring. The intermediate mathematics of functions is planned to be discussed in Calculus and advanced part in functional analysis. Please note that I am regularly revising older articles and trying to maintain the accuracy and completeness. If you feel that there is any fault or incompleteness in a post then please make a comment on respective post. If you are interested in writing a guest article on this blog, then kindly email me at mdnb[at]live[dot]in.