Problem 2 We say that a binary relation R S Sx S on a set S satisfies the diamond property if the following condition holds: A binary relation R on S is said to satisfy confluence if the reflexive and transitive closureR* of the binary relation R (i.e., R* is the least relation such that (i) R S R*, (i) V E S. (Rx), (iii) Vx, y, z E S. (zR"?? yR"z-> xR2)) satisfies the diamond property. A binary relation R on S is said to satisfy weak confluence if the following condition holds: For example, R1 = {(a, b), (a,e), (b, d), (c, e), (d, e)) satisfies confluence and weak confluence, but (1) Give an example of a binary relation on the set (a, b, c, d) that satisfies weak confluence but (2) Prove that, for every set S and every binary relation R on S, if R satisfies the diamond (3) Prove that, for every set S and every binary relation R on S, if R satisfies weak confluence (4) Prove that, for every binary relation R on the set fa, b, c, if R satisfies weak confluence, R does not satisfy the diamond property. Answer the following questions. not confluence. property, then R also satisfies confluence and also if there is no infinite sequence xoRxiRx2R.. ., then R satisfies confluence. also satisfies confluence Problem 2 We say that a binary relation R S Sx S on a set S satisfies the diamond property if the following condition holds: A binary relation R on S is said to satisfy confluence if the reflexive and transitive closureR* of the binary relation R (i.e., R* is the least relation such that (i) R S R*, (i) V E S. (Rx), (iii) Vx, y, z E S. (zR"?? yR"z-> xR2)) satisfies the diamond property. A binary relation R on S is said to satisfy weak confluence if the following condition holds: For example, R1 = {(a, b), (a,e), (b, d), (c, e), (d, e)) satisfies confluence and weak confluence, but (1) Give an example of a binary relation on the set (a, b, c, d) that satisfies weak confluence but (2) Prove that, for every set S and every binary relation R on S, if R satisfies the diamond (3) Prove that, for every set S and every binary relation R on S, if R satisfies weak confluence (4) Prove that, for every binary relation R on the set fa, b, c, if R satisfies weak confluence, R does not satisfy the diamond property. Answer the following questions. not confluence. property, then R also satisfies confluence and also if there is no infinite sequence xoRxiRx2R.. ., then R satisfies confluence. also satisfies confluence