1. Let T = {a, b, c, d}.
a. Find an example of a relation of this set that is transitive, but neither reflexive nor symmetric.
For this one I had { (a,b) (b,c) (a,c) }. I am not sure if this is correct.
b. Find an example of a relation on this set that is both transitive and symmetric, but not reflexive.
For this one I came up with { (a, b) (b, c) (a, c) (b, a) (c, b) (c, a) }. Is this correct?
a. Your example { (a, b) (b, c) (a, c) } is indeed transitive, meaning that if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. However, this example is also reflexive because it includes (a, a), (b, b), and (c, c), which satisfy the reflexive property. So, it is not an example of a relation that is transitive but neither reflexive nor symmetric.
To find an example that is transitive but neither reflexive nor symmetric, you would need to identify a relation that does not contain elements of the form (a, a), (b, b), (c, c), or (d, d), and also a relation that contains pairs like (a, b), (b, c), and (a, c), but not the pairs (b, a) or (c, a). One possible example could be:
R = { (a, b), (b, c), (a, c), (d, b), (c, d) }
This relation is transitive because whenever (a, b) and (b, c) are in the relation, it also includes (a, c). However, it is not reflexive because it does not contain elements like (a, a), (b, b), (c, c), or (d, d). Additionally, it is not symmetric because it contains the pair (d, b) but not (b, d).
b. Your example { (a, b), (b, c), (a, c), (b, a), (c, b), (c, a) } is indeed a relation that is transitive and symmetric, but not reflexive. This relation satisfies the transitive property because whenever (a, b) and (b, c) are in the relation, it also includes (a, c). It satisfies the symmetric property because for every pair (x, y) in the relation, it also includes the pair (y, x). However, it is not reflexive because it does not contain elements like (a, a), (b, b), (c, c), or (d, d).
Well done on coming up with the correct examples!