Question

R1 = {(a,
b.∈ r2 | a > b}, the "greater than" relation, r2 = {(a,
b.∈ r2 | a ≥ b}, the "greater than or equal to" relation, r3 = {(a,
b.∈ r2 | a < b}, the "less than" relation, r4 = {(a,
b.∈ r2 | a ≤ b}, the "less than or equal to" relation, r5 = {(a,
b.∈ r2 | a = b}, the "equal to" relation, r6 = {(a,
b.∈ r2 | a ≠ b}, 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.

Answer 1

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 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 < b or b > 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 < b or b > 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