3. Assessment:

The following questions were given to Chat GPT.

Determine what's correct, fix all the mistakes, and assign a letter grade: A,B,C,D,F.

Let A be an arbitrary set. Let R be a relation such that RCAXA

Prove or disprove the following:

a) If R is both symmetric and reflexive, then R is transitive.

1. If R is both symmetric and reflexive, it does not necessarily imply that R is transitive. Here's

a counterexample:

Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (1, 2), (2, 1)). In this case, R is reflexive because every

element in A is related to itself. R is also symmetric because if (a, b) is in R, then (b, a) is also

in R. However, R is not transitive because (1, 2) and (2, 1) are in R, but (1, 1) is not in R. This

violates the transitive property./nb) If A is non-empty and Vr € A, 3y = A Ray but R is both anti-symmetric and

symmetric, then R is reflexive.

The statement is false.

Counterexample:

Let A = {1, 2} and R = {(1, 2), (2, 1)).

In this case, A is non-empty and every element in A relates to another element in A since for

every element a in A, there exists an element b in A such that (a, b) is in R.

R is symmetric because for every pair (a, b) in R, the pair (b, a) is also in R.

R is also antisymmetric because if (a, b) and (b, a) are in R, then a = b. In this case, (1, 2) and

(2, 1) are in R, and we have 1 # 2, so the antisymmetric property holds.

However, R is not reflexive because (1, 1) and (2, 2) are not in R.

Therefore, the given statement is false, as we have found a counterexample that satisfies the

conditions of non-empty A, every element in A relating to another element in A, and R being

both antisymmetric and symmetric, but R is not reflexive.

c) There exists an A, R such that |A| = 2 and R is transitive but not reflexive.

2. There exists an A with cardinality two such that R is transitive but not reflexive. Let A = {1, 2}

and R = {(1,2)). In this case, R is transitive because there is only one pair (1, 2), and there are

no pairs that violate the transitive property. However, R is not reflexive because (1, 1) and (2,

2) are not in R.

Therefore, the statement is true.