Question

For each of the following relations on the set {0, 1, 2, 3}. • R1 = {(0, 0),(1, 1),(2, 2),(3, 3)} • R2 = {(1, 1),(2, 2),(3, 3)} • R3 = {(0, 0),(1, 1),(2, 0),(2, 2),(2, 3),(3, 3)} • R4 = {(0, 0),(0, 1),(1, 0),(1, 1),(2, 2),(3, 3)} (a) Which of these relations are reflexive? (b) Which of these are symmetric? (c) Which of these are anti-symmetric? (d) Which of these are transitive?

Answer 1

(a)
`R_1, R_3, R_4`

(b)
`R_1, R_2, R_4`

(c)
`R_1, R_2, R_3`

(d)
`R_4`

Explanation:

Since,

A relation R defined on A is called reflexive if,

∀ x ∈ A, (x, x)∈ R

It is called symmetric,

∀ x, y ∈ A, if (x, y)∈ R then (x, y)∈ R such that x ≠ y,

It is called anti symmetric,

∀ x, y ∈ A, if (x, y)∈ R then (x, y)∈ R such that x = y,

It is called transitive,

∀ x, y, z ∈ A, if (x, y)∈ R and (y, z)∈ R then (x, z)∈ R such that x ≠ y,

Given set,

{0, 1, 2, 3},

Also, the relation defined on the set are,

•
`R_1` = {(0, 0),(1, 1),(2, 2),(3, 3)}

•
`R_2` = {(1, 1),(2, 2),(3, 3)}

•
`R_3` = {(0, 0),(1, 1),(2, 0),(2, 2),(2, 3),(3, 3)}

•
`R_4` = {(0, 0),(0, 1),(1, 0),(1, 1),(2, 2),(3, 3)}

Hence, by the above explanation it is clear that,

Relations which are reflexive,

`R_1, R_3, R_4`

Relations which are symmetric,

`R_1, R_2, R_4`

Relations which are anti-symmetric,

`R_1, R_2, R_3`

Relations which are transitive,

`R_4`