Question

Refer to the relation R on the set {1,2,3,4,5) defined by the rule (x, y) = R if 3 divides x - y

1. List the elements of R

2. List the elements of R-1

3. Find the domain of R

4. Find the range of R

5. Find the domain of R-1

6. Find the range of R-1

Give examples of relations on {1,2,3,4} having the properties specified in the following:

10. Reflexive, antisymmetric, and not transitive

9. Not reflexive, not symmetric, and transitive

8. Reflexive, not symmetric, and not transitive

7. Reflexive, symmetric, and not transitive

Answer 1

R = {(1,1), (2,2), (3,3), (4,4), (5,5), (1,4), (2,5), (4,1), (5,2)}

We can receive $R^{-1}$ by inverse all of the pairs (x, y) in R to (y, x)

$R^{-1}$ = {(1,1), (2,2), (3,3), (4,4), (5,5), (4,1), (5,2), (1,4), (2,5)}

Domain of R and $R^{-1}$ is the set of all x in the pairs (x, y)

Range of R and $R^{-1}$ is the set of all y in the pairs (x, y)

DomR = {1, 2, 3, 4, 5}

Range R= {1, 2, 3, 4, 5}

$DomR^{-1}=$ {1, 2, 3, 4, 5}

$RangeR^{-1}=$ {1, 2, 3, 4, 5}

Reflexive relation R on the set X - such a relation, for which $\forall x\in X: (xRx)$

Symmetric relation R on the set X - such a relation, for which $\forall x,y \in R: xRy\to yRx$

Transitive relation R on the set X - such a relation, for which $\forall x,y,z \in X: (xRy \land yRz)\to xRz$

Antireflexive relation R on the set X - such a relation, for which $\forall x\in X: \lnot(xRx)$

Antisymmetric relation is such a relation for which if condition of the symmetry is true then x = y

10. Reflexive, antisymmetric, and not transitive

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

9. Not reflexive, not symmetric, and transitive

R = {(1,2), (2,3), (1,3)}

8. Reflexive, not symmetric, and not transitive

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

7. Reflexive, symmetric, and not transitive

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