Binary Relations

Introduction of binary relations:-

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms dyadic relation and 2-place relation are synonyms for binary relations.(Source: Wikipedia).

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Example problems for binary relation:-


Problem 1:-

Find four binary relations from `{a,b}` to `{x,y}` that are not functions from `{a,b}` to `{x,y}`

Solution:-


R = `{(a, x), (a, y), (b, x)} ` is not a function from `{a, b}` to `{x, y}` because two values related to a
R = `{(a, x)} ` is not a function from `{a, b}` to `{x, y}` because there are no values related to b

It able to find another two easily. In fact, of the 16 binary relations from `{a, b}` to `{x, y}` , only 4 are functions, and 12 are not functions.



Problem 2:-

Find four binary relations from `{a,b,c}` to `{x,y,z}` that are not functions from `{a,b,c}` to `{x,y,z}`

Solution:-

R = `{(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z)} `  is not a function from` {a,b,c}` to `{x,y,z}` because three values related to a.

R = `{(a, x)}` is not a function from `{a, b, c}` to `{x, y,z}` because there are no values related to b & c.


It able to find another two easily. In fact, of the 36 binary relations from `{a, b, c}` to `{x, y, z}` , only 6 are functions, and 30 are not functions. Please express your views of this topic What is Rational Number by commenting on blog.


More example problems for binary relation:-


Problem 1:-

Do you understand the following calculations?

X = `{1,2,3}`

|x| = 3.

Solution:-

X x X = `{(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)}`

| X x X | = 9

|P(X x X)| = `2^9`
Because a binary relation on X is any subset of pairs in X x X there are `2^9` possible binary relations on X.
(Some authors say` 2^9 -1` because of not liking an empty relation.)



Problem 2:-

Do you understand the following calculations?

X = `{1,2,3,4}`

|x| = 4.

Solution:-

X x X = `{(1,1),(1,2),(1,3),(1,4)(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)}`

| X x X | = 16

|P(X x X)| = `2^16`
Because a binary relation on X is any subset of pairs in X x X there are` 2^16` possible binary relations on X.
(Some authors say `2^16 -1` because of not liking an empty relation.)