Mathematics
Equivalence relation
Practice questions
For real numbers, (aRb) if (a\le b). Why is this not an equivalence relation?For real numbers, (aRb) if (a<b). Why is this not an equivalence relation?For real numbers, (aRb) if both (a) and (b) are positive. Why is this not an equivalence relation?For natural numbers, (aRb) if (a) and (b) have the same last digit. What type of relation is it?If (R) is an equivalence relation, which statement is always true?On (A={1,2,3,4}), how many equivalence classes are formed by the same parity relation?On the set of integers, (aRb) if (|a|=|b|). What type of relation is it?The set (A={1,2,3,4,5}) is divided into classes ({1,4},{2},{3,5}). Relation (R) means belonging to the same class. Which pair must be in (R)?On (A={1,2,3}), why is (R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}) not an equivalence relation?In (A={1,2,3,4,5,6,7,8}), (aRb) if (a\equiv b \pmod{4}). What is the equivalence class of (4)?On the set (A={1,2,3}), the relation (R={(1,1),(2,2),(3,3)}) is given. What type of relation is it?Which three properties are required for a relation on a set to be called an equivalence relation?On (A={1,2,3}), (R={(1,1),(2,2),(3,3),(1,2),(2,1)}). Is it an equivalence relation?If (R) is defined on integers by (aRb) when (a-b) is even, then what type of relation is (R)?On (A={1,2,3,4}), (aRb) means (a) and (b) are both even or both odd. Choose the correct statement about (R).On the set (A={1,2,3}), the universal relation (R=A\times A) is given. What type of relation is it?For a non-empty set (A), which statement is correct about the equality relation (R={(a,a):a\in A})?If (R) is an equivalence relation and ((a,b)\in R), then which pair must belong to (R) due to the symmetric property?If (R) is an equivalence relation and ((a,b)\in R), ((b,c)\in R), what follows by the transitive property?On (A={1,2,3}), (R={(1,1),(2,2)}). Why is it not an equivalence relation?