Rosen(relations)

Question

Given the directed graphs representing two relations, how can the directed graph of the union, intersection, symmetric difference, difference, and composition of these relations be found?

Comments

"representing two relations" or "Representing relations between two sets"???

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Make ordered pairs for the directed graphs representing two relations.

Then from those ordered pairs for two relations you can easily find union, intersection, symmetric difference, difference, and composition of these relations.

example : Assuming  directed graph for R1 and R2 ordered pairs for both are written.You can check the directed graph from R1 and R2 by plotting them.

                R1 = {(1, 1), (1, 3), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (4, 1)}

                R2 = {(1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 3), (4, 1), (4, 3)}.

               R1 U R2 ={(1, 3) , (2, 1) , (2, 3) , (3, 1) , (4, 1) }

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there are two different directed graphs representing two different relations...

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@bikram sir, is there any way to directly find the graph of composite relation by seeing the graphs of individual relations, without making the ordered pairs?

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@shraddha

Yes, we can find the graph of composite relation .

Let A and B are the domain and co domain of the function f.

It means that every element x of A has an image f (x) in B.

we can understand composition in terms of two functions.
Let there be two functions denoted as :

Observe that set B is common to two functions. The rules of the functions are given by f (x) and g (x) respectively. Our objective here is to define a new function  and its rule.

For every element x in A, there exists an element f (x) in set B. This is the requirement of function f by definition. For every element f (x) in B, there exists an element g(f (x)) in set B. This is the requirement of function g by definition. It follows, then, that for every element xin A, there exists an
element g(f(x)) in set C.

This concluding statement is definition of a new function :

By convention, we call this new function as  and is read “g composed with f“.

Function composition is a special relation between sets not common to two functions.

The two symbolical representations are equivalent.

see the graph