Question

About Here are two relations defined on the set {a, b, c, d}: S = { (a, b), (a, c), (c, d), (c, a) } R = { (b, c), (c, b), (a, d), (d, b) } Write each relation as a set of ordered pairs. (a) S ο R (b) R ο S (c) S ο S

Answer 1

Answer with Step-by-step explanation:

We are given that a set {a,b,c,d}

S={(a,b),(a,c),(c,d),(c,a)}

R={(b,c),(c,b),(a,d),(d,b)]

Composition of relation:Let R and S are two relations on the given set

If ordered pair (a,b) belongs to relation R and (b,c) belongs to S .

Then, SoR={(a,c)}

By using this rule

SoR={(b,d),(b,a)}[/tex]

Because
`(b,c)\in R` and
`(c,d)\in S`.Thus,
`(b,d)\in SoR`

`(b,c))\in R` and
`(c,a)\in S`.Thus,
`(b,a)\in SoR`

b.RoS={(a,c),(a,b),(c,b),(c,d)}

Because

`(a,b)\in S,(b,c)\in R` .Therefore, the ordered pair
`(a,c)\in` RoS

`(a,c)\in S,(c,b)\in R` .Thus,
`(a,b)\in RoS`

`(c,d)\in S,(d,b)\in R`.Thus,
`(c,b)\in RoS`

`(c,a)\in S,(a,d)\in R`.Thus,
`(c,d)\in RoS`

c.SoS={(a,d),(a,a),(c,c),(c,b)}

Because

`(a,c)\;and\; (c,d)\in S`.Thus,
`(a,d)\in SoS`

`(c,a),(a,b)\in S`.Thus,
`(c,b)\in SoS`

`(a,c)\in S` and
`(c,a)\in S`.Thus,
`(a,a)\in SoS`

`(c,a)\in S` and
`(a,c)\in S`.Thus ,
`(c,c)\in SoS`