INVERSE OF A RELATION
Consider once again the example M is the set of men, and W the set of women and C Ì M ´ W where (x, y) Î C if x is the husband of y. So, the relation “is the husband of “ then determines a second relation, namely “is the wife of”.
Consequently, from C = {(x, y): x Î M, y Î W and x is the husband of y},
We can obtain the new relation
C –1 = {(y, x): y Î W, x Î M and y is the wife of x}.
The relation C –1 is the inverse of the relation C.
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In general, ... Let C be a relation between A and B. The inverse relation of C is the relation C –1 Ì B ´ A defined by C –1 = {(y, x): (x, y) Î C} Example: Let A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {(1, 2), (2, 4), (3, 6), (4, 8)}.
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1. If g = { (x, y): x Î Â and y = 2x}, then what is g –1 ?
2. Let Z = { x : x Î Z } be the set of all integers and define the relation C Ì Z ´ Z by C ={(x, y): (x, y) Î Z ´ Z and x – y is divisible by 3}. Determine if the following statements are true or false.
a. For each x Î Z, (x, x) Î C
b. If x Î Z, and y = x / 3, then (x, y) Î C.
c. If (x, y) Î C, then (y, x) Î C.
d. If (x, y) Î C and (y, z) Î C, then (x, z) Î C
e. C –1 = C
