Construct R3, S2 and S º 3, by supposing R and S are relations on {a,b,c,d), where R={(a,b),(a,d),(b,c),(c,c),(d,a)} and S = {(a,c), (b,d)(d,a)}.

To construct R3, S2, and S º 3, we need to perform various operations on the given relations R and S.

1. R3: R3 refers to the composition of relation R with itself three times. To compute R3, we need to find all ordered pairs that appear after composing R with itself three times.

To get R2:
R2 = R º R = {(x,y) | ∃z (xRz ∧ zRy)}

1. Start with the first element in R and check if there is a second element we can reach through a common element.
- (a, b) → (b, c) (c, c) (c, d) (d, a)
- (a, d) → (d, a) (a, b) (a, d)
- (b, c) → (c, c)
- (c, c) → (c, c)
- (d, a) → (a, b) (a, d)

2. Combine all the results to get R2:
R2 = {(a, b), (a, d), (b, c), (c, c), (d, a)}

To get R3:
R3 = R2 º R = {(x, y) | ∃z (xR2z ∧ zR)}

1. Start with the first element in R2 and check if there is a second element we can reach through a common element.
- (a, b) → (b, c) (c, c) (c, d) (d, a)
- (a, d) → (d, a) (a, b) (a, d)
- (b, c) → (c, c)
- (c, c) → (c, c)
- (d, a) → (a, b) (a, d)

2. Combine all the results to get R3:
R3 = {(a, b), (a, d), (b, c), (c, c), (d, a)}

2. S2: S2 refers to the composition of relation S with itself two times. We need to find all ordered pairs that appear after composing S with itself twice.

To get S2:
S2 = S º S = {(x, y) | ∃z (xSz ∧ zSy)}

1. Start with the first element in S and check if there is a second element we can reach through a common element.
- (a, c) → (c, a) (a, d)
- (b, d) → (d, a) (a, c)
- (d, a) → (a, c) (c, a)

2. Combine all the results to get S2:
S2 = {(a, c), (b, d), (d, a)}

3. S º 3: S º 3 refers to the composition of relation S with relation R3.

To get S º 3:
S º 3 = S º R3 = {(x, y) | ∃z (xSz ∧ zR3y)}

1. Start with the first element in S and check if there is a second element we can reach through a common element in R3.
- (a, c) → (c, c) (c, a) (a, b) (a, d)
- (b, d) → (d, a) (a, b) (a, d)
- (d, a) → (a, b) (a, d)

2. Combine all the results to get S º 3:
S º 3 = {(a, b), (a, d), (b, d), (d, a)}

So, R3 = {(a, b), (a, d), (b, c), (c, c), (d, a)}
S2 = {(a, c), (b, d), (d, a)}
S º 3 = {(a, b), (a, d), (b, d), (d, a)}