Mathematics
Equivalence relation
Practice questions
If (R) is an equivalence relation on a (7)-element set and (R) has (19) ordered pairs. Which class-size pattern is possible?If (A) has (5) elements and (R) is an equivalence relation with (13) pairs, what can be the class sizes?On (A={1,2,3,4,5,6}), (aRb) holds when (a) and (b) have the same remainder on division by (7). What is this relation?On (A={1,2,3,4,5}), (aRb) holds when (a+b) is divisible by (6) or (a=b). Why is this an equivalence relation?If (R) is an equivalence relation and ([a]\subseteq B\subseteq A), but (B) is not an equivalence class. Which reason is possible?On (A={1,2,3,4,5,6}), relation (R) has classes ({1,2,6},{3,5},{4}). Along with ((6,1)) and ((5,3)), which pair must be in (R)?On integers, (aRb) holds when (a-b) is even, and (aSc) holds when (a-c) is divisible by (3). How many equivalence classes are there in (R\cap S)?On (A={1,2,3,4,5}), (aRb) holds when (|a-b|\leq 1). Why is it not an equivalence relation?On (A={1,2,3,4,5,6}), (aRb) holds when (a) and (b) have the same remainder modulo (3). What type of relation is the complement of (R)?On (A={1,2,3,4,5,6,7}), (aRb) holds when (a) and (b) have the same divisibility status by (3) and the same divisibility status by (2). How many ordered pairs are in this relation?On (A={1,2,3,4,5,6}), (aRb) holds when (a^2+b^2) is even. What are the equivalence classes of this relation?On integers, (aRb) holds when (a^2\equiv b^2 \pmod{9}). Which integer must belong to the equivalence class of (4)?On (A={1,2,3,4,5,6,7,8,9}), (aRb) holds when (\gcd(a,9)=\gcd(b,9)). How many ordered pairs are in the relation?On real numbers, (aRb) holds when (a^2+2a=b^2+2b). Which is the equivalence class of (-4)?On (A={1,2,3,4,5,6,7,8}), (aRb) holds when (\min(a,5)=\min(b,5)). Which is the equivalence class of (7)?If (A) has (6) elements and an equivalence relation has exactly (31) ordered pairs, what can be the class sizes?On (A={1,2,3,4,5,6}), (aRb) holds when (a) and (b) have the same remainders modulo (2) and modulo (5). Which is the equivalence class of (3)?On integers, (aRb) holds when (a-b) is divisible by (8) or by (12). Why is this relation not an equivalence relation?On the set of all subsets of (S={1,2,3,4,5,6}), (A R B) holds when (|A|=|B|). How many members are in the equivalence class of a (2)-element subset?On (A={1,2,3,4,5,6,7,8}), (aRb) holds when (a) and (b) have the same divisibility status by (4) and by (2). How many equivalence classes are formed?