Mathematics
Equivalence relation
Practice questions
On integers, (aRb) holds when (a^2+a=b^2+b). Which is the equivalence class of (2)?On (A={1,2,3,4,5,6,7,8}), (aRb) holds when (\gcd(a,8)=\gcd(b,8)). Which is the equivalence class of (6)?On (A={1,2,3,4,5,6,7,8}), how many ordered pairs are in the relation defined by (\gcd(a,8)=\gcd(b,8))?On (A={1,2,3,4,5}), the classes of relation (R) are ({1,4,5}) and ({2,3}). Which pair will definitely not be in (R)?If an equivalence relation on a (6)-element set has class sizes (3,2,1), how many pairs are in the relation?How many equivalence relations on (A={1,2,3,4,5}) have (1,2,3) in the same class?How many equivalence relations on (A={1,2,3,4,5}) have exactly three equivalence classes?How many equivalence relations on (A={1,2,3,4}) have (1) and (2) in different classes?On (A={1,2,3,4,5,6}), (aRb) holds when (a) and (b) have the same remainder on division by (5). How many pairs are in the relation?On integers, (aRb) holds when (a-b) is divisible by both (6) and (10). This is the same as congruence modulo which number?On integers, (aRb) holds when (a-b) is divisible by (4) or by (6). Why is this relation not an equivalence relation?On real numbers, (aRb) holds when (\sin a=\sin b). What type of relation is it?For the relation (\sin a=\sin b) on real numbers, which is the correct form of the class of (0)?On (A={1,2,3,4,5,6,7,8,9}), (aRb) holds when (a) and (b) have the same number of prime factors counted with repetition. Which is the equivalence class of (6)?On (A={1,2,3,4,5,6}), (aRb) holds when (a) and (b) have the same remainder modulo (2) and also the same remainder modulo (3). Which is the equivalence class of (5)?On (A={1,2,3,4,5,6,7,8,9,10}), (aRb) holds when (a) and (b) have the same greatest common divisor with (10). How many equivalence classes are formed?On (A={1,2,3,4,5,6,7,8,9,10}), for the relation (\gcd(a,10)=\gcd(b,10)), which is the equivalence class of (8)?On (A=\mathbb{R}), (aRb) holds when (\lfloor a\rfloor=\lfloor b\rfloor). Which is the equivalence class of (-1.2)?If (R) is an equivalence relation and (aRb), (bRc), (cRd), which conclusion must be true?If (R) is an equivalence relation and ([a]\cap[b]\cap[c]\neq\varnothing), which statement is correct?