Which of the following statements are true?
I. Every relation that is symmetric and transitive is reflexive.
II. The identity relation on a set is an equivalence relation.
III. The universal relation on a set is an equivalence relation.
IV. A relation can be both symmetric and not reflexive.

II, III and IV onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerII and III onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I is false: $\{(1, 1)\}$ on $\{1, 2\}$ is symmetric and transitive but lacks $(2, 2)$ .
II and III are true (both relate each element to itself, and are symmetric and transitive). IV is true: $\{(1, 2), (2, 1)\}$ .

Worth noting

The 'proof' that symmetric and transitive give reflexive assumes every element is related to something.

Concept: Facts about the three properties
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsTypes of relations

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Ten questions from Relations & Functions, at your own pace.

Which of the following statements are true?I. Every relation that is symmetric and transitive is reflexive.II. The identity relation on a set is an equivalence relation.III. The universal relation on a set is an equivalence relation.IV. A relation can be both symmetric and not reflexive.

Explanation

Which of the following statements are true?
I. Every relation that is symmetric and transitive is reflexive.
II. The identity relation on a set is an equivalence relation.
III. The universal relation on a set is an equivalence relation.
IV. A relation can be both symmetric and not reflexive.