Question

Briefly discuss the Relations on Sets, give examples.

2. Briefly discuss introduce Reflexivity, Symmetry, and Transitivity, give examples

Answer 1

ANSWER:

Relation:

A relation is a relationship between sets of values.The relation is between the X-values and y-values of orderd pairs in math.

-> The set of x-value is called the Domain,nad the sat of all y-values is called the range.

-> Relations can be displayed as a table, a mapping or a graph. In a table the x-values and y-values are listed in separate columns.

-> A binary relation R from set x to y is a subset of the Cartesian product x × y.

-> If there are two sets A and B, and relation R have order pair (x, y).

a) The domain of R is the set { x | (x, y) ∈ R for some y in B }

b) The Range of R is the set { y| (x,y) ∈ R for some x in A }

Example:

let x={1,2,8} and y={1,3,7}

a) if relation r is equal to then R={(1,1),(3,3)}

b) If relation R is less than then R={(1,3),(1,7),(2,3),(2,7)}

c) If relation R is greater than R={(2,1),(9,1)(9,3),(9,7)}

Types of Relations:

1) The Empty Relation between sets X and Y, or on E, is the empty set ∅.

2) The Full Relation between sets X and Y is the set X × Y.

3) The Identity Relation on set X is the set {(x, x) | x ∈ X}.

4) The Inverse Relation R' of a relation R is defined as − R’ = {(b, a) | (a, b) ∈ R}

Example :

If R = {(1, 2), (2, 3)} then R’ will be {(2, 1), (3, 2)}

5) A relation R on set A is called Reflexive if ∀a∈A is related to a (aRa holds).

Example :

The relation R = {(a, a), (b, b)} on set X = {a, b} is reflexive.

6) A relation R on set A is called Irreflexive if no a ∈ A is related to a (aRa does not hold).

Example :

The relation R = {(a, b), (b, a)} on set X = {a, b} is irreflexive.

2)Reflexivity, Symmetry, and Transitivity:

a) Reflexivity: R is reflexive if for all x Î A, (x,x) Î R.

example:

The relation R on {1,2,3} given by R = {(1,1), (2,2), (2,3), (3,3)} is reflexive. (All loops are present.).

b) Symmetric:

R is symmetric if for all x,y Î A, (x,y) Î R implies (y,x) Î R. (Equivalently, for all x,y Î A, x R y implies that y R x.)

example:

The relation R on {1,2,3} given by R = {(1,1), (1,2), (2,1), (1,3), (3,1)} is symmetric. (All paths are 2-way.)

c)Transitive:

R is transitive if for all x,y,z Î A, (x,y) Î R and (y,z) Î R implies (x,z) Î R. (Equivalently, for all x,y,z Î A, x R y and y R z implies x R z.)

Example:

The relation R on {1,2,3} given by R = {(1,1), (1,2), (2,1), (2,2), (2,3), (1,3)} is transitive.