Mathematics
Equivalence relation
Practice questions
On natural numbers, (aRb) if and only if (a) and (b) have the same set of prime factors. Which of the following belongs to the equivalence class of (12)?If two equivalence classes (B_1) and (B_2) of an equivalence relation satisfy (B_1\cap B_2\neq \varnothing), what conclusion follows?On (A={1,2,3,4}), (R=A\times A). Which statement about its equivalence classes is correct?On (A={1,2,3,4,5,6}), (aRb) if and only if (a) and (b) are both multiples of (3) or both not multiples of (3). What is the equivalence class of (3)?On real numbers, (aRb) if and only if (a-b\in Z). Which is the correct form of the equivalence class of (1.7)?On (A={1,2,3,4,5}), (aRb) if and only if (\min(a,3)=\min(b,3)). What is the equivalence class of (5)?Let (R) be an equivalence relation on a set (A). If ([a]\neq[b]), which of the following is definitely true?On (A={1,2,3,4,5,6,7}), (aRb) if and only if (a) and (b) leave the same remainder on division by (2). Which statement about ((4,7)) is correct?On the set of all students, (xRy) if and only if (x) and (y) have the same date of birth. Why is this an equivalence relation?On (A={1,2,3,4}), which is the smallest equivalence relation?On (A={1,2,3,4,5,6,7,8}), (aRb) if and only if (a-b) is divisible by (4). Which is the equivalence class of (3)?On integers, (aRb) if and only if (a^2-b^2) is divisible by (8). Which of the following pairs belongs to the relation?On real numbers, (aRb) if and only if (a-b\in Q). Which is the equivalence class of (\sqrt{2})?On (A={1,2,3,4,5}), the relation is (R={(a,b):a\equiv b \pmod{2}}). How many ordered pairs are in this relation?A partition of (A={1,2,3,4,5,6}) is ({{1,2,6},{3,5},{4}}). Choose the correct statement about ((2,6)) and ((3,4)) in the equivalence relation formed by it.On real numbers, (aRb) if and only if (a^2=b^2). Which is the equivalence class of (-5)?On all non-zero real numbers, (aRb) if and only if (\frac{a}{b}\in Q). What type of relation is this?On the set of all real ordered pairs, ((a,b)R(c,d)) if and only if (a+b=c+d). How can the equivalence class of ((2,5)) be written?On (A={1,2,3,4}), (R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(2,3),(3,2)}) is given. Which minimum pairs must be added to make it an equivalence relation?If (A) has (4) elements, how many different equivalence relations are possible on (A)?