1. Let R = {(1, y), (1, z), (3,y)} be a relation from A = {1, 2,3} to B = {x, y, z}.
a. Find $R^{-1}$
b. Compare $(R^{-1})^{-1}$ and R.
2. Show that the relation
f = {(1, a), (2, b), (3,a)}
a. Defines a function from A = {1, 2, 3} to B = {a, b, c}. Find its range.
b. Show that the relation f = {(1, a), (2, b), (3, c), (1,b)} does not define a function from A = {1, 2,3} to B = {a, b, c}.
3. Given the following two relations from A={1, 2,4} to B = {2, 6, 8,10}:
aRb if and only if a|b. aSb if and only if b - 4 = a.
List the elements of R, S, R\cup S, and $R \cap S$.
4. Let
R = {(1, 2), (1, 6), (2, 4), (3, 4), (3, 6), (3, 8)}
S = {(2, u), (4, s), (4, t), (6, t), (8,u)}
Find $S \circ R$.
5. Let A = {1, 2}, B = {1}. Show that A × B. = B × A.
6. Let A = {1, 2, 3, 4}, R = {(1, 2), (2, 4), (3, 1)}, and S = {(1, 1), (2, 3), (4, 3)}. Then find:
a) $R^{-1}$
b) $S^{-1}$
c) $S \circ R$
d) $R \circ S$
7. a) Let A = {2, 3, 4} and B = { 3, 4, 5, 6, 7 }. Define the relation R by aRb if and only if a divides b. Find, R, Dom(R), Range(R).
b) Let A = {1, 2, 3, 4}. Define the relation R by aRb if and only if a ? b. Find, R, Dom(R), Range(R).
