Question

Let X={a,b,c,d},

f={(a,b),(b,a),(c,c),(d,d)}⊂X×X, and g={(a,d),(b,a),(c,b),(d,c)}⊂X×X.
Find g∘g. a. {(a,a),(b,d),(c,b),(d,c)}
b. {(a,d),(b,b),(c,a),(d,c)}
c. {(a,a),(b,b),(c,c),(d,d)}
d. {(a,c),(b,d),(c,a),(d,b)

Answer 1

To compute g∘g, apply the function g to each element and then apply g again to the resulting value. The resulting set for g∘g is {(a, c), (b, d), (c, a), (d, b)}, corresponding to answer option d.

Step-by-step explanation:

To find g∘g we must apply the function g to itself. This means we'll use the output of g as the input for the second application of g. Here is a step-by-step explanation:

1. Start with the first pair (a, d) in g. The result of applying g to 'a' is 'd'.
2. We then apply g to 'd', and as per the definition of g, 'd' maps to 'c'. Therefore, g(g(a)) = c.
3. Repeat this process for each element in the set X:
4. g(g(b)) maps 'b' to 'a' and then 'a' to 'd', so we get (b, d).
5. g(g(c)) maps 'c' to 'b' and then 'b' to 'a', so we get (c, a).
6. g(g(d)) maps 'd' to 'c' and then 'c' to 'b', so we get (d, b).

Collecting all the resulting pairs, the set for g∘g is {(a, c), (b, d), (c, a), (d, b)}. Therefore, the correct answer is option d.