Mathematics
Equivalence relation
Practice questions
On (A={1,2,3,4}), (R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}). Is (R) an equivalence relation?On integers, (aRb) if (|a-b|\leq 2). Why is this not an equivalence relation?On (A=\mathbb{R}), (xRy) if (x-y=0). Which relation is this?If an equivalence relation on (A={1,2,3}) contains ((1,2)) and ((2,3)), what is the minimum number of pairs in (R)?On (A={1,2,3,4,5,6}), (aRb) if (a) and (b) leave the same remainder on division by (3). What is ([2])?On (A=\mathbb{Z}), (aRb) if (a-b) is divisible by (4). How many distinct equivalence classes does this relation have?An equivalence relation (R) on a set (A) has classes ({a,b}), ({c}), and ({d,e,f}). How many pairs are in (R)?On real numbers, (xRy) if (x^3=y^3). What is the correct conclusion about this relation?On (A=\mathbb{R}), (xRy) if (x-y\geq 0). Why is this not an equivalence relation?On (A={1,2,3,4,5,6}), (aRb) if (a+b) is divisible by (3). Why is this not an equivalence relation?On (A={1,2,3,4}), an equivalence relation has the single class ({1,2,3,4}). Which relation is it?If two equivalence classes ([a]) and ([b]) of an equivalence relation are distinct, what is true about their intersection?On natural numbers, (aRb) if (a) and (b) have the same sum of digits. What type of relation is this?On (A={1,2,3,4,5,6,7,8}), (aRb) if (a) and (b) lie in the same block among ({1,2}), ({3,4,5}), ({6,7,8}). How many pairs are in the relation?On (A=\mathbb{Z}), (aRb) if (a-b) is even. What is true about ([1]) and ([2])?On (A={1,2,3,4,5}), (aRb) if (a) and (b) have the same remainder on division by (2). Which pair will not be in (R)?If (A) has (5) elements and (R) is the equality relation, how many pairs are in (R)?If (A) has (5) elements and (R=A\times A), how many equivalence classes will (R) have?On (A=\mathbb{R}), (xRy) if (\sin x=\sin y). What type of relation is this?On (A={1,2,3,4,5,6}), relation (R) has classes ({1,6}), ({2,3,5}), and ({4}). Which pair definitely will not be in (R)?