Mathematics
Equivalence relation
Practice questions
On all real numbers, (aRb) if and only if (a^2-a=b^2-b). Which is the equivalence class of (3)?On all real numbers, (aRb) if and only if (a^2+2a=b^2+2b). Which is the equivalence class of (-1)?On all real numbers, (aRb) if and only if (\sin^2 a=\sin^2 b). Which of the following numbers belongs to the equivalence class of (0)?On all real numbers, (aRb) if and only if (\cos^2 a=\cos^2 b). Which of the following must be in the equivalence class of (\frac{\pi}{3})?On all positive real numbers, (aRb) if and only if (\frac{a}{b}) is an integral power of (2). Which is the equivalence class of (3)?On positive integers, (aRb) if and only if (a) and (b) have the same power of (2), that is, (a=2^k m) and (b=2^k n), where (m,n) are odd. What forms the equivalence class of (12)?If (R) and (S) are equivalence relations on a set (A), how can the equivalence classes of (R\cap S) be understood?If (A={1,2,3}), (R) has classes ({1,2},{3}), and (S) has classes ({1},{2,3}), why is (R\cup S) not an equivalence relation?On a set (A), which is the largest equivalence relation?On a set (A), which is the smallest equivalence relation?On the set (A={1,2,3,4,5,6}), relation (R) is defined by ((a,b)\in R) if (a-b) is divisible by (3). How many equivalence classes does this relation have?If (R={(a,b):a+b\text{ is even}}) is defined on the set of integers, which statement about (R) is correct?On (A={1,2,3,4}), relation (R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,4),(4,3)}) is given. What is the equivalence class of (1)?On (A={1,2,3}), (R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}). Why is this not an equivalence relation?On real numbers, relation (R) is defined by ((x,y)\in R) if (x-y\in \mathbb{Z}). Which description is correct?On (A={1,2,3,4,5,6,7,8}), (aRb) if (a\equiv b \pmod{4}). What is the equivalence class of (5)?If (R) is an equivalence relation on a set (A) and ([a]\cap[b]\neq \varnothing), which conclusion is definitely true?A partition of (A={1,2,3,4,5}) is ({{1,3,5},{2,4}}). How many ordered pairs will be in the equivalence relation formed by this partition?On (A={1,2,3,4}), the equivalence classes are ({1,4}) and ({2,3}). Which pair must belong to the relation?On natural numbers, (aRb) if (a) and (b) have the same number of prime factors, counted with repetition. What type of relation is this?