Mathematics
Equivalence relation
Practice questions
On (A={1,2,3,4,5,6,7,8,9}), (aRb) holds when (a) and (b) have the same greatest common divisor with (9). Which is the equivalence class of (6)?For points in the plane, ((x_1,y_1)R(x_2,y_2)) holds when (x_1+y_1=x_2+y_2). Which is the equivalence class of ((2,-1))?For points in the plane, ((x_1,y_1)R(x_2,y_2)) holds when (x_1-y_1=x_2-y_2). What form do the equivalence classes have?If (R) is an equivalence relation and (aRb), what is ([a]\setminus[b])?If (R) is an equivalence relation and ([a]\setminus[b]\neq\varnothing), which conclusion is correct?On (A={1,2,3,4,5,6,7}), relation (R) has classes ({1,7},{2,6},{3,5},{4}). How many pairs are in (R)?On (A={1,2,3,4,5,6,7}), (aRb) holds when (a+b=8) or (a=b). What type of relation is this?On integers, (aRb) holds when (a-b) is divisible by (7). What is ([3]\cap[10])?On integers, (aRb) holds when (a-b) is divisible by (7). What is ([3]\cap[11])?On real numbers, (aRb) holds when (a-b) is rational. Which statement is correct?On non-zero real numbers, (aRb) holds when (\frac{a}{b}\in\mathbb{Q}). If (aRb) and (bRc), which expression is useful to prove transitivity?On (A={1,2,3,4,5,6,7,8,9,10,11,12}), (aRb) holds when (a) and (b) have the same remainder on division by (6). How many pairs are in (R)?If (A) has (4) elements, how many equivalence relations have no two distinct elements in the same class?If (A) has (4) elements, how many equivalence relations have all elements in one class?On (A={1,2,3,4,5}), (R) has classes ({1,2},{3},{4,5}). How many pairs are in the complement of (R)?On (A={1,2,3,4,5}), (R) has classes ({1,2,3},{4,5}). Why is the complement of (R) not an equivalence relation?On (A={1,2,3,4}), for (R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}), what is the equivalence class of (4)?In which situation is (R\cup S), where (R) and (S) are equivalence relations, definitely an equivalence relation?If an equivalence relation has (2) classes on a (7)-element set, what is the minimum possible number of pairs in the relation?If an equivalence relation has (2) classes on a (7)-element set, what is the maximum possible number of pairs in the relation?