Question 1 (1 point) Which of the following statements are true? Select all that apply. The relation ? on the set of integers is an equivalence relation. Every relation is an equivalence relation. The graph of a function f : A ? A is a relation on A. If ~ is an equivalence relation on some set, then any two equivalences classes of ~ are either equal or disjoint. Question 2 (1 point) Consider the relation ~ on Z given by x ~ y ? x - y is even. True or false: ~ is an equivalence relation. True False
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The relation on the set of integers is an equivalence relation: This statement is incomplete, as it does not specify which relation is being referred to. So, we cannot determine if it is true or false. Show more…
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(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (c) Statement- 1 is True, Statement- 2 is False (d) Statement- 1 is False, Statement- 2 is True Statement 1 A relation $\mathrm{R}$ is defined on the set of real numbers. $x R y$ if $x-y$ is positive. Then, $R$ is neither reflexive nor symmetric. and Statement $\underline{2}$ A relation $\mathrm{R}$ on a set $\mathrm{A}$ is reflexive if $(\mathrm{x}, \mathrm{x}) \in \mathrm{R}$ for all $\mathrm{x} \in \mathrm{A}$ and $\mathrm{R}$ is symmetric if $(\mathrm{x}, \mathrm{y}) \in \mathrm{R}$ implies $(\mathrm{y}, \mathrm{x}) \in \mathrm{R}$.
Consider the following relation on the set of real numbers: R = {(x, y) | x - y is an integer} a) Prove that this is an equivalence relation. b) What is the equivalence class of 1 for this equivalence relation? c) What is the equivalence class of 1/2 for this equivalence relation?
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Let R be a relation on the set of positive integers with xRy <=> xy =1 which of the following statements are NOT true about the relation R? Select one or more. A. R is transitive B. R is anti-symmetric C. R is symmetric D. R is reflexive
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