Mathematics
Equivalence relation
Practice questions
If (R) and (S) are equivalence relations on (A), the classes of (R\cap S) are obtained from what?On (A={1,2,3,4,5,6}), (R) has classes ({1,2,3},{4,5,6}), and (S) has classes ({1,4},{2,5},{3,6}). How many classes are in (R\cap S)?On integers, (aRb) holds when (a\equiv b \pmod{4}), and (aSb) holds when (a\equiv b \pmod{6}). In (R\cap S), what is the equivalence class of (14)?On (A={1,2,3,4,5}), relation (R) contains all diagonal pairs, ((1,2),(2,1),(2,5),(5,2)). Which minimum pairs must be added to make it an equivalence relation?On (A={1,2,3,4,5,6}), (aRb) holds when (a+b) is divisible by (7) or (a=b). What are the classes of this relation?On (A={1,2,3,4,5,6}), for the relation where (a+b) is divisible by (7) or (a=b), how many ordered pairs are there?On real numbers, (aRb) holds when (\cos a=\cos b). Which is the correct form of the equivalence class of (0)?On (A={1,2,3,4,5,6,7,8,9,10}), (aRb) holds when (\tau(a)=\tau(b)), where (\tau(n)) is the number of positive divisors. Which is the equivalence class of (6)?On (A={1,2,3,4,5,6,7,8,9,10}), how many equivalence classes are formed by the relation (\tau(a)=\tau(b))?If (R) is an equivalence relation and ([a]\subseteq [b]\cup[c]), with ([a]\cap[b]\neq\varnothing), which conclusion is correct?On (A={1,2,3,4,5,6,7,8}), (aRb) holds when (a) and (b) have the same status of divisibility by (3) and by (4). Which is the equivalence class of (8)?On real numbers, (aRb) holds when (|a-1|=|b-1|). Which is the equivalence class of (-2)?On (A={1,2,3,4,5,6,7,8,9}), (aRb) holds when the digit sums of (a) and (b) give the same remainder modulo (3). Which is the equivalence class of (8)?If (A) has (5) elements, how many equivalence relations are possible on (A)?How many equivalence relations on (A={1,2,3,4,5}) have (1) alone in its own class?How many equivalence relations on (A={1,2,3,4,5}) have (1) and (2) in the same class, but (3) not in that class?If (R) and (S) are equivalence relations on (A) and (R\subseteq S), which statement is always true?On (A={1,2,3,4,5,6}), relation (R) has classes ({1,2,3},{4,5},{6}). How many ordered pairs are not in (R)?On integers, (aRb) holds when (a-b) is divisible by (5), and (aSc) holds when (a-c) is divisible by (10). What is (R\cap S) equal to?On the integers, aRb when a-b is divisible by 5, and aSc when a-c is divisible by 10. What is R∪S equal to?