Please explain why these proofs are incorrect and provide a correct answer: 1/ Let x, y ∈ Z. Prove that if xy and x + y

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Please explain why these proofs are incorrect and provide acorrect answer:
1/ Let x, y ∈ Z. Prove that if xy and x + y are even, thenx and y are both even.
Proof: By definition, x + y = 2k and x*y = 2h. So, x = (2h)/y, x+ (2h)/y = 2k.
So both are numbers are even as an even plus even is even
2/ A relation R is defined on Z by xRy ifx3 − y ≡ 0 (mod 3). Prove or disprove: R isreflexive, R is symmetric, and R is transitive
Proof:
Reflexive: Note that x3 ≡ y (mod 3). Since,
13 ≡ 1 (mod 3)
23 ≡ 2 (mod 3)
33 ≡ 3 (mod 3)
the relation is reflexive
Symmetric: The relation can not be symmetric because63−2 ≡ 216−2 ≡ 214 ≡ 1 (mod 3), but 233 − 6 ≡8 − 6 ≡ 2 (mod 3).
Transitive: Can’t be transitive for the same reason as it can’tbe symmetric. It is reflexive, so (1, 1) ∈ R and (1, 2) ̸∈ R. Thus,(2, 1) ̸∈ R