On R - 0, the relation R is a R b if ab > 0. Which of these describes R?

Question 29

MediumOriginal question

On $\mathbb{R} - \{0\}$ , the relation $R$ is $a \mathrel{R} b$ if $ab > 0$ . 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

$a \cdot a = a^{2} > 0$ for $a \ne 0$ , and $ab = ba$ : reflexive and symmetric.
$ab > 0$ means $a$ and $b$ have the same sign, and 'same sign' passes along a chain: transitive.

Worth noting

With 0 removed, nothing can link a negative number to a positive one.

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

The rule this testsTypes of relations

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On R−{0}\mathbb{R} - \{0\}R−{0}, the relation RRR is aRba \mathrel{R} baRb if ab>0ab > 0ab>0. Which of these describes RRR?

Explanation

On $\mathbb{R} - \{0\}$ , the relation $R$ is $a \mathrel{R} b$ if $ab > 0$ . Which of these describes $R$ ?