Mathematics
Equivalence relation
Practice questions
On real numbers, (aRb) if and only if (\lfloor a\rfloor=\lfloor b\rfloor). Which is the equivalence class of (-1.2)?On all (2\times 2) real matrices, (A RB) if and only if (\operatorname{tr}(A)=\operatorname{tr}(B)). Why is this an equivalence relation?On the set of all non-zero polynomials, (pRq) if and only if (\deg(p)=\deg(q)). What forms the equivalence class of (x^2+1)?On (A={1,2,3,4,5,6}), (aRb) if and only if (\gcd(a,b)>1). Is this an equivalence relation?On real numbers, (aRb) if and only if (|a-b|<1). Is this an equivalence relation?On (A={1,2,3,4,5}), (aRb) if and only if (a\le b). Why is this not an equivalence relation?On all complex numbers, (z_1Rz_2) if and only if (|z_1|=|z_2|). What does the equivalence class of (3+4i) represent?On (A={1,2,3,4,5,6,7,8,9,10}), (aRb) if and only if (a) and (b) are both prime or both not prime. Which is the equivalence class of (4)?On the set of all linear equations (ax+by+c=0), two equations are related if they represent the same line. What type of relation is this?On natural numbers, (aRb) if and only if (a) and (b) have the same number of digits. What forms the equivalence class of (125)?On (A={1,2,3,4,5,6}), (aRb) if and only if (\max(a,b)) is even. Is this an equivalence relation?On (A={1,2,3,4}), (R=A\times A). How many equivalence classes does it have?On (A={1,2,3,4}), the identity relation is (I_A={(1,1),(2,2),(3,3),(4,4)}). How many equivalence classes does it have?If (R) is an equivalence relation and ([a]\cap[b]\neq\varnothing), which of the following is always true?On (A={1,2,3,4,5,6}), (aRb) if and only if the greatest common divisor of (a) and (6) equals the greatest common divisor of (b) and (6). Which is the equivalence class of (5)?On all real numbers, (aRb) if and only if (e^a=e^b). What will the equivalence classes be like?On real numbers, (aRb) if and only if (\cos a=\cos b). Which of the following numbers must be in the equivalence class of (0)?On (A={1,2,3,4,5,6,7,8}), (aRb) if and only if (a) and (b) are of the same type: both less than (4) or both greater than or equal to (4). How many ordered pairs are there?On the set of all vectors in (R^2), (uRv) if and only if (|u|=|v|). Which is the equivalence class of the zero vector?On all integers, (aRb) if and only if (a) and (b) leave the same remainder on division by (10). Choose the correct statement about (27) and (-13).