40 RELATIONS [CHAP. 2
SupplementaryProblems
RELATIONS
2.20. LetS={a, b, c},T={b, c, d}, andW={a, d}. FindS×T×W.
2.21. Findxandywhere: (a)(x+ 2 , 4 )=( 5 , 2 x+y); (b)(y− 2 , 2 x+ 1 )=(x− 1 ,y+ 2 ).
2.22. Prove: (a)A×(B∩C)=(A×B)∩(A×C); (b)A×(B∪C)=(A×B)∪(A×C).
2.23. Consider the relationR={( 1 , 3 ), ( 1 , 4 ),( 3 , 2 ), ( 3 , 3 ), ( 3 , 4 )}onA={ 1 , 2 , 3 , 4 }.(a) Find the matrixMRofR. (d) Draw the directed graph ofR.
(b) Find the domain and range ofR. (e) Find the composition relationR◦R.
(c) FindR−^1. (f) FindR◦R−^1 andR−^1 ◦R.2.24. LetA={ 1 , 2 , 3 , 4 },B={a, b, c},C={x, y, z}. Consider the relationsRfromAtoBandSfromBtoCas follows:R={( 1 , b), ( 3 , a), ( 3 , b), ( 4 ,c)} and S={(a, y), (c, x), (a, z)}(a) Draw the diagrams ofRandS.
(b) Find the matrix of each relationR,S(composition)R◦S.
(c) WriteR−^1 and the compositionR◦Sas sets of ordered pairs.2.25. LetRandSbe the following relations onB={a, b, c, d}:R={(a, a), (a, c), (c, b), (c, d), (d, b)} and S={(b, a), (c, c), (c, d), (d, a)}Find the following composition relations: (a)R◦S; (b)S◦R; (c)R◦R; (d)S◦S.
2.26. LetRbe the relation onNdefined byx+ 3 y=12, i.e.R={(x, y)|x+ 3 y= 12 }.(a) WriteRas a set of ordered pairs. (c) FindR−^1.
(b) Find the domain and range ofR. (d) Find the composition relationR◦R.PROPERTIES OF RELATIONS
2.27. Each of the following defines a relation on the positive integersN:(1) “xis greater thany.” (3) x+y= 10
(2) “xyis the square of an integer.” (4)x+ 4 y=10.Determine which of the relations are: (a) reflexive; (b) symmetric; (c) antisymmetric; (d) transitive.
2.28. LetR andSbe relations on a setA. AssumingAhas at least three elements, state whether each of the
following statements is true or false. If it is false, give a counterexample on the setA={ 1 , 2 , 3 }:(a) IfRandSare symmetric thenR∩Sis symmetric.
(b) IfRandSare symmetric thenR∪Sis symmetric.
(c) IfRandSare reflexive thenR∩Sis reflexive.