Mathematics
Equivalence relation
Practice questions
On (A={1,2,3,4,5,6,7,8,9,10,11,12}), (aRb) if and only if (a) and (b) leave the same remainder on division by (6). Which is the equivalence class of (5)?On integers, (aRb) if and only if (a-b) is divisible by (7). How will the equivalence class of (-15) be written?On real numbers, (aRb) if and only if (a-b\in Z). Which is the equivalence class of (\frac{3}{4})?On (A={1,2,3,4,5,6,7}), (aRb) if and only if (a^2\equiv b^2 \pmod{7}). Which is the equivalence class of (3)?On all real numbers, (aRb) if and only if (\lfloor 2a\rfloor=\lfloor 2b\rfloor). Which is the equivalence class of (\frac{5}{4})?On real numbers, (aRb) if and only if (\lfloor 2a\rfloor=\lfloor 2b\rfloor). Which is the correct equivalence class of (\frac{5}{4})?On (A={1,2,3,4,5,6}), (aRb) if and only if (\operatorname{lcm}(a,6)=\operatorname{lcm}(b,6)). Which is the equivalence class of (4)?On all (3\times 3) real matrices, (A RB) if and only if (\operatorname{rank}(A)=\operatorname{rank}(B)). How many equivalence classes are formed?On all real numbers, (aRb) if and only if (a^4=b^4). Which is the equivalence class of (2)?On all complex numbers, (z_1Rz_2) if and only if (\operatorname{Re}(z_1)=\operatorname{Re}(z_2)). What does the equivalence class of (2+3i) represent?On (A={1,2,3,4,5,6,7,8}), (aRb) if and only if (\lceil \frac{a}{2}\rceil=\lceil \frac{b}{2}\rceil). Which is the equivalence class of (5)?On all real numbers, (aRb) if and only if (a-b\in Q). Which of the following pairs is not in the relation?On the set of all polynomials, (pRq) if and only if (p(1)=q(1)). By which condition is the equivalence class of (p(x)=x^2+x) formed?On (A={1,2,3,4,5,6}), (R={(a,b):a) and (b) leave the same remainder on division by (3)(}). How many ordered pairs are in (R)?On a set with (8) elements, an equivalence relation has class sizes (2,2,4). How many ordered pairs are in the relation?A partition ({{a,c,d},{b}}) is given on (A={a,b,c,d}). Which pair will not belong to the equivalence relation formed from it?On positive real numbers, (aRb) if and only if (\frac{\ln a}{\ln b}) is defined and rational. Why is this not an equivalence relation?On all real numbers, (aRb) if and only if (a^2+b^2=0). Is this an equivalence relation?On (A={1,2,3,4}), (aRb) if and only if (a+b) is prime. Why is this not an equivalence relation?On (A={1,2,3,4,5}), (aRb) if and only if (|a-b|) is divisible by (3). What are the equivalence classes of this relation?