Which of these relations are equivalence relations? I. On the students of a school, 'is in the same class as'. II. On R, 'a - b is an integer'. III. On Z, 'a ≤ b + 1'. IV. On the triangles in a plane, 'is similar to'.

Question 68

HardOriginal question

Which of these relations are equivalence relations?
I. On the students of a school, 'is in the same class as'.
II. On $\mathbb{R}$ , ' $a - b$ is an integer'.
III. On $\mathbb{Z}$ , ' $a \le b + 1$ '.
IV. On the triangles in a plane, 'is similar to'.

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

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

I, II and IV are reflexive, symmetric and transitive; for II, differences of integers are integers.
III is not symmetric: $0 \le 5 + 1$ but $5 \le 0 + 1$ is false.

Worth noting

II looks unusual, but adding two integer differences gives an integer difference.

Concept: Recognising equivalence relations
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsEquivalence relations and classes

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

Which of these relations are equivalence relations?I. On the students of a school, 'is in the same class as'.II. On R\mathbb{R}R, 'a−ba - ba−b is an integer'.III. On Z\mathbb{Z}Z, 'a≤b+1a \le b + 1a≤b+1'.IV. On the triangles in a plane, 'is similar to'.

Explanation

Which of these relations are equivalence relations?
I. On the students of a school, 'is in the same class as'.
II. On $\mathbb{R}$ , ' $a - b$ is an integer'.
III. On $\mathbb{Z}$ , ' $a \le b + 1$ '.
IV. On the triangles in a plane, 'is similar to'.