Which of these relations on 0 1 2 3 are partial orderings If

Which of these relations on {0, 1, 2, 3} are partial orderings? If not, state all the properties it lacks.

(1) {(0,0), (1,1), (2,2), (3,3), (2,3), (3,2)}

(2) {(0,0), (1,1), (2,2), (3,3), (2,3), (1,3), (2,0)}

A relation is partial order if it is REFLEXIVE, ANTI-SYMMETRIC, and TRANSITIVE.

{0, 1, 2, 3}

1. {(0,0), (1,1), (2,2), (3,3), (2,3), (3,2)}

This relation is not partial order since it is not staisying anti-symmetirc properties.

A relation is anti-symmetirc if (a,b) is in R and (b,a) is R then a=b.

(2,3) is in R and (3,2) is R so as per anti-symmetric relation 2=3 which is not true.

(2) {(0,0), (1,1), (2,2), (3,3), (2,3), (1,3), (2,0)}

This relation is paartial oreder relation since

all elements are related to itself --- reflexive

there is no element such that (a,b) is R and (b,a) is R and a is not equal to b - Anti0Symmetric

(2,0) and (0,0) is R so (2,0) is in R

(2,3) and (3,3) is R so (2,3) is in R

(2,2) and (2,3) is R so (2,3) is in R

(2,2) and (2,0) is R so (2,0) is in R

(1,1) and (1,3) is R so (1,3) is in R

therefor is is transitive also

Thus it is partial order,