R1 = (a,

b.∈ r2, the "greater than" relation, r2 = a ≥ b, the "greater than or equal to" relation, r3 = (a,

b.∈ r2, the "less than" relation, r4 = a ≤ b, the "less than or equal to" relation, r5 = a = b, the "equal to" relation, r6 = (a,

b.∈ r2, the "unequal to" relation. find

a. r2 ◦ r1.

b. r2 ◦ r2.

c. r3 ◦ r5.

d. r4 ◦ r1.

e. r5 ◦ r3. f) r3 ◦ r6. g) r4 ◦ r6. h) r6 ◦ r6.

For relations X and Y, (x, y) belongs to X o Y if and only if there exists z such that (x, z) belongs to X and (z, y) belongs to Y.

Part A:

(a, b) belongs to r2 ◦ r1 if there exists c such that (a, c) belong to r2 and (c, b) belongs to r1.

(a, c) belongs to r2 means that a ≥ c and (c, b) belongs to r1 means that c > b.

a ≥ c and c > b means that a ≥ c > b which means that a > b which belong to r1.

Therefore, r2 ◦ r1 = r1

Part B:

(a, b) belongs to r2 ◦ r2 if there exists c such that (a, c) belong to r2 and (c, b) belongs to r2.

(a, c) belongs to r2 means that a ≥ c and (c, b) belongs to r2 means that c ≥ b.

a ≥ c and c ≥ b means that a ≥ b which belong to r2.

Therefore, r2 ◦ r2 = r2

Part C:

(a, b) belongs to r3 ◦ r5 if there exists c such that (a, c) belong to r3 and (c, b) belongs to r5.

(a, c) belongs to r3 means that a < c and (c, b) belongs to r5 means that c = b.

a < c and c = b means that a < b which belong to r3.

Therefore, r3 ◦ r5 = r3

Part D:

(a, b) belongs to r4 ◦ r1 if there exists c such that (a, c) belong to r4 and (c, b) belongs to r1.

(a, c) belongs to r4 means that a ≤ c and (c, b) belongs to r1 means that c > b.

a ≤ c and c > b means that a b or a = b which means a ≠ b or a = b which belong to r5 ∪ r6.

Therefore, r4 ◦ r1 = r5 ∪ r6

Part E:

(a, b) belongs to r5 ◦ r3 if there exists c such that (a, c) belong to r5 and (c, b) belongs to r3.

(a, c) belongs to r5 means that a = c and (c, b) belongs to r3 means that c < b.

a = c and c < b means that a < b which belong to r3.

Therefore, r5 ◦ r3 = r3

Part F:

(a, b) belongs to r3 ◦ r6 if there exists c such that (a, c) belong to r3 and (c, b) belongs to r6.

(a, c) belongs to r3 means that a < c and (c, b) belongs to r6 means that c ≠ b

a < c and c ≠ b means that a a or a = b which means that a ≠ b or a = b which belong to r5 ∪ r6.

Therefore, r3 ◦ r6 = r5 ∪ r6.

Part G:

(a, b) belongs to r4 ◦ r6 if there exists c such that (a, c) belong to r4 and (c, b) belongs to r6.

(a, c) belongs to r4 means that a ≤ c and (c, b) belongs to r6 means that c ≠ b

a ≤ c and c ≠ b means that a a or a = b which means that a ≠ b or a = b which belong to r5 ∪ r6.

Therefore, r3 ◦ r6 = r5 ∪ r6.

Part H:

(a, b) belongs to r6 ◦ r6 if there exists c such that (a, c) belong to r6 and (c, b) belongs to r6.

(a, c) belongs to r6 means that a ≠ c and (c, b) belongs to r6 means that c ≠ b.

a ≠ c and c ≠ b means that a = b or a ≠ b which belong to r5 ∪ r6.

Therefore, r2 ◦ r2 = r5 ∪ r6