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

Question 20

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

Reflexive and symmetric, but not transitiveCorrect answerYour answerNeither reflexive, symmetric nor transitiveCorrect answerYour answerSymmetric onlyCorrect answerYour answerSymmetric and transitive, but not reflexiveCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$a + a = 2a$ is even, so no number is related to itself: not reflexive. $a + b = b + a$ , so it is symmetric.
1 and 2 are related, 2 and 3 are related, but $1 + 3 = 4$ is even: not transitive.

Worth noting

An odd sum pairs an odd number with an even one, so a number is never related to itself.

Concept: Testing a relation given by a rule
Typical timeTypical: 90 s
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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 odd. Which of these describes RRR?

Explanation

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