need help with the numbers 4.3 and 4.4 below. I included pictures.

4.3: 2, 7, 15, 17, 22

Transcribed Image Text:

THEOREM 2 Let R be a relation on a set A. Then 4.4 Exercises In Exercises 1 through 8, let A = {1, 2, 3, 4]. Determine whether the relation is reflexive, irreflexive, symmetric, asym- metric, antisymmetric, or transitive. 5. R (a) Reflexivity of R means that a = R(a) for all a in A. (b) Symmetry of R means that a = R(b) if and only if b = R(a). (c) Transitivity of R means that if b = R(a) and c = R(b), then c = R(a). 1. R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4,3), (4,4)} 2. R = {(1,2), (1, 3), (1, 4), (2, 3), (2, 4), (3,4)} 3. R= {(1,3), (1, 1), (3, 1), (1, 2), (3, 3), (4,4)} 4. R = {(1, 1), (2, 2), (3, 3)) 6. R= A x A 7. R = {(1,2), (1, 3), (3, 1), (1, 1), (3, 3), (3, 2), (1,4), (4, 2), (3, 4)) 8. R= {(1,3), (4, 2), (2, 4), (3, 1), (2, 2)) In Exercises 9 and 10 (Figures 4.22 and 4.23), let A = {1, 2, 3, 4,5). Determine whether the relation R whose digraph is given is reflexive, irreflexive, symmetric, asymmetric, antisym- metric, or transitive. 11. 0 1 0 1 1 0 1 1 010 0 1 100 10. Figure 4.22 In Exercises 11 and 12, let A = {1, 2, 3, 4). Determine whether the relation R whose matrix MR is given is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transi- tive. 12. Figure 4.23 100 0 0 00 1 0 0001 2 In Exercises 13 through 24, determine whether the relation R on the set A is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. 13. A = Z; a R b if and only if a b+ 1. 14. A = Z+; a R b if and only if la - b| 2. THEOREM 2 Let R be a relation on a set A. Then 4.4 Exercises In Exercises 1 through 8, let A = {1, 2, 3, 4]. Determine whether the relation is reflexive, irreflexive, symmetric, asym- metric, antisymmetric, or transitive. 5. R (a) Reflexivity of R means that a = R(a) for all a in A. (b) Symmetry of R means that a = R(b) if and only if b = R(a). (c) Transitivity of R means that if b = R(a) and c = R(b), then c = R(a). 1. R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4,3), (4,4)} 2. R = {(1,2), (1, 3), (1, 4), (2, 3), (2, 4), (3,4)} 3. R= {(1,3), (1, 1), (3, 1), (1, 2), (3, 3), (4,4)} 4. R = {(1, 1), (2, 2), (3, 3)) 6. R= A x A 7. R = {(1,2), (1, 3), (3, 1), (1, 1), (3, 3), (3, 2), (1,4), (4, 2), (3, 4)) 8. R= {(1,3), (4, 2), (2, 4), (3, 1), (2, 2)) In Exercises 9 and 10 (Figures 4.22 and 4.23), let A = {1, 2, 3, 4,5). Determine whether the relation R whose digraph is given is reflexive, irreflexive, symmetric, asymmetric, antisym- metric, or transitive. 11. 0 1 0 1 1 0 1 1 010 0 1 100 10. Figure 4.22 In Exercises 11 and 12, let A = {1, 2, 3, 4). Determine whether the relation R whose matrix MR is given is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transi- tive. 12. Figure 4.23 100 0 0 00 1 0 0001 2 In Exercises 13 through 24, determine whether the relation R on the set A is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. 13. A = Z; a R b if and only if a b+ 1. 14. A = Z+; a R b if and only if la - b| 2.