Mathematics
Equivalence relation
Practice questions
On the set (A={1,2,3,4}), the relation (R={(a,b):a-b\text{ is divisible by }2}) is given. Choose the correct statement about this relation.On the set of integers (Z), (aRb) if and only if (a-b) is divisible by (5). Which is the equivalence class of (7)?On the set of real numbers, (aRb) if and only if (|a|=|b|). What is the equivalence class of (-3) for this relation?On the set (A={1,2,3,4,5,6}), (aRb) if and only if (\gcd(a,6)=\gcd(b,6)). How many elements are in the equivalence class of (4)?On the set of all straight lines in a plane, (lRm) if and only if (l) and (m) have the same slope. What type of relation is this?On the set of (2\times 2) real matrices, (A RB) if and only if (\det(A)=\det(B)). Why is this relation an equivalence relation?On the set of all functions from (A={1,2,3}) to (A), (fRg) if and only if (f(1)=g(1)). Choose the correct statement about this relation.On the set (A={1,2,3,4,5}), (aRb) if and only if (a) and (b) are both even or both odd. How many blocks are formed in the partition by this relation?On integers, (aRb) if and only if (a+b) is even. Which of the following statements is correct?On the set of non-zero real numbers, (aRb) if and only if (\frac{a}{b}>0). On what basis is this an equivalence relation?On (A={1,2,3}), (R={(1,1),(2,2),(3,3),(1,2),(2,1)}) is given. Which minimum pair must be added to make it an equivalence relation?On (A={1,2,3}), (R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)}) is given. Which pairs are definitely needed to make it an equivalence relation?A partition of (A={1,2,3,4}) is ({{1,4},{2},{3}}). How many ordered pairs will be in the equivalence relation formed by it?An equivalence relation on a set (A) has equivalence classes ({a,b,c}), ({d,e}), and ({f}). How many ordered pairs are there in this relation?On (A={1,2,3,4}), (aRb) if and only if (a^2-b^2) is divisible by (3). What is the equivalence class of (1)?On real numbers, (aRb) if and only if (a-b) is rational. Choose the correct statement about the equivalence classes of this relation.On (A={1,2,3,4,5,6}), (aRb) if and only if (a\equiv b \pmod{3}). What are the distinct equivalence classes of this relation?On the set of lowercase English alphabet letters, (aRb) if and only if (a) and (b) are both vowels or both consonants. How many equivalence classes does this relation form?On (A={1,2,3,4}), (R={(1,1),(2,2),(3,3),(4,4),(1,3),(3,1)}). Which partition is obtained from this relation?On natural numbers, (aRb) if and only if (a) and (b) have the same unit digit. Which statement about this relation is correct?