Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 423 — #431
10.11. Summary of Relational Properties 423
Next to each of the relational formulas, write the roman numerals of all the predicate
formulas equivalent to it. It is not necessary to name the property expressed,
but you can get partial credit if you do. For example, part (a) gets the label “i.” It
expresses irreflexivity.
i. 8d: NOT.d R d/
ii. 8d:
d R d
iii. 8c; d:
iv. 8c; d:
c R d IFF d R c
c R d IMPLIES d R c
v. 8c; d: c R d IMPLIES NOT.d R c/
vi. 8c ¤ d: c R d IMPLIES NOT.d R c/
vii. 8c ¤ d: c R d IFF NOT.d R c/
viii. 8b; c; d:
.b R c AND c R d/ IMPLIES b R d
ix. 8b; d:
Œ9c: .b R c AND c R d/ IMPLIES b R d
x. 8b; d: b R d IMPLIES Œ9c: .b R c AND c R d/
(a) R \ Id D D ; i.
(b) R R 1
(c) R D R 1
(d) Id D R
(e) R ı R R
(f) R R ı R
(g) R \ R 1 Id D
(h) R R 1
(i) R \ Id R D R 1 \ Id R
(j) R \ R 1 D ;