Which of these relations on{0,1,2,3}are partial orderings? Determine the properties of a partial ordering that the others lack. a) { (0,0), (2,2), (3,3) } b) { (0,0), (1,1), (2,0), (2,2), (2,3), (3,3) } c) { (0,0), (1,1), (1,2), (2,2), (3,1), (3,3) } d) { (0,0), (1,1), (1,2), (1,3), (2,0), (2,2), (2,3), (3,0), (3,3) } e) { (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3), (2,0), (2,2), (3,3)

Step-by-step explanation:

A = {0,1,2,3}

a) : R = { (0,0), (2,2), (3,3) }

R is antisymmetric, because if the (a, b) ∈R, than a=b.

R is not reflexive, because (1,1) ∉ R while 1 ∈ A.

R is transitive, because if the (a, b) ∈R and (b, c) ∈ R, than a=b=c and (a, c) = (a, a) ∈R.

R is not portable ordering because R is not reflexive.

b) : R = { (0,0), (1,1), (2,0), (2,2), (2,3), (3,3) }

R is antisymmetric, because if the (a, b) ∈R and if the (b, a) ∈ R, than a=b (since (2,0) ∈ R and (0,2) ∉ R; and (2,3) ∈ R and (3,2) ∉ R)

R is reflexive, because (a, a) ∈ R of every element a ∈ A.

R is transitive, because if the (a, b) ∈R and if the (b, c) ∈R. then a = b or b = c (since there are only two element not of the form (a, a) and that pair does not satisfy (a, b) ∈ R and (b, a) ∈ R), which implies (a, c) = (b, c) ∈ R or (a, c) = (a, b) ∈ R.

R is a partial ordering, because R is reflexive, antisymmetric and transitive.

c) : R = { (0,0), (1,1), (1,2), (2,2), (3,1), (3,3) }

R is reflexive, because (a, a) ∈R of every element a ∈ A.

R is antisymmetric, because if the (a, b) ∈R and if the (b, a) ∈R. then a = b (since (1, 2) ∈R and (2, 1) ∉ R; (3, 1) ∈ R and (1, 3) ∉ R).

R is not transitive, because (3, 1) ∈ R and (1, 2) ∈R, while (3, 2) ∈ R.

R is not a partial ordering. because R is not transitive.

d) : R = { (0,0), (1,1), (1,2), (1,3), (2,0), (2,2), (2,3), (3,0), (3,3) }

R is the reflexive, because (a, a) ∈R of every elements∈A.

R is the antisymmetric, because if the (a, b) ∈R and if the (b, a) ∈R, then a = b (since (1. 2) ∈R and (2. 1) ∉R; similarly, all other elements not of the form (a, a)).

R is not transitive, because (1, 2) ∈R and (2, 0) ∈R, while (1. 0) ∉R.

R is not a partial ordering, because R is not transitive,

e) : R = { (0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 3) }

R is the reflexive, because (a, a) ∈R of every element a∈A.

R is not antisymmetric, because (1, 0) ∈R and (0, 1) ∈R while 0 is not equal to 1.

R is not transitive, because (2, 0) ∈Rand (0, 3) ∈R, while (2, 3) ∉R.

R is not a partial ordering, because R is not the antisymmetric and not the transitive.