R is the relation on Z given by a R b if a + b is even. Which of these describes R?

Question 22

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$R$ is the relation on $\mathbb{Z}$ given by $a \mathrel{R} b$ if $a + b$ is even. Which of these describes $R$ ?

Reflexive and symmetric, but not transitiveCorrect answerYour answerReflexive, symmetric and transitiveCorrect answerYour answerSymmetric and transitive, but not reflexiveCorrect answerYour answerReflexive and transitive, but not symmetricCorrect answerYour answer

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CorrectNot quiteAnswer B

Correct answer: B

$a + a = 2a$ is even, and $a + b = b + a$ : reflexive and symmetric.
An even sum means $a$ and $b$ have the same parity; same parity is passed on, so it is transitive.

Worth noting

Even sum means the same parity, and 'same parity' is an equivalence.

Concept: Testing a relation given by a rule
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsTypes of relations

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RRR is the relation on Z\mathbb{Z}Z given by aRba \mathrel{R} baRb if a+ba + ba+b is even. Which of these describes RRR?

Explanation

$R$ is the relation on $\mathbb{Z}$ given by $a \mathrel{R} b$ if $a + b$ is even. Which of these describes $R$ ?