Mathematics
Equivalence relation
Practice questions
On (A={1,2,3,4,5,6}), (aRb) if (\gcd(a,6)=\gcd(b,6)). What is the equivalence class of (4)?If (R) and (S) are both equivalence relations on a set (A), which statement about (R\cap S) is correct?On integers, (aRb) if (a^2\equiv b^2 \pmod{5}). What type of relation is this?On (A=\mathbb{R}\setminus{0}), (xRy) if (\frac{x}{y}>0). How many equivalence classes does this relation have?On (A={1,2,3,4,5,6}), (aRb) if (a) and (b) leave the same remainder when divided by (2). How many ordered pairs are in the relation?A relation (R) on a set (A) is reflexive and symmetric but not transitive. Which conclusion about (R) is correct?On (A={1,2,3,4,5}), one equivalence class of (R) is ({1,2,5}). If (R) is an equivalence relation, which pair cannot be in (R)?On real numbers, (xRy) if (x-y) is rational. Which statement about (R) is correct?On (A={1,2,3,4,5,6,7}), (aRb) if (a-b) is divisible by (2). What are the equivalence classes of (R)?On (A=\mathbb{Z}), (aRb) if (a-b) is divisible by (6). What is the equivalence class of (0)?For a non-empty set (A), which statement about the universal relation (A\times A) is correct?For a non-empty set (A), which statement about the empty relation (\varnothing) is correct?On (A={1,2,3,4}), how many distinct equivalence relations are possible with exactly two equivalence classes?If (f:A\to B) is a function and (aRb) if (f(a)=f(b)), what type of relation is (R)?On (A={-2,-1,0,1,2}), (xRy) if (x^2=y^2). What is the equivalence class of (2)?On (A=\mathbb{R}), (xRy) if (|x|=|y|). What type of relation is this?On integers, (aRb) if (a+b) is divisible by (2). Which statement is correct?On (A={1,2,3,4,5,6}), (aRb) if (\lfloor\frac{a}{2}\rfloor=\lfloor\frac{b}{2}\rfloor). Which are the equivalence classes of (R)?If (R) is an equivalence relation and ((a,b)\in R), which statement is necessarily true?On (A={1,2,3,4,5,6,7,8,9}), (aRb) if (a\equiv b \pmod{3}). How many ordered pairs are in the relation?