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 ,$(c,c)\in SoS$