Question 5. Let ~ be the equivalence relation defined on Z by x~ y 3 divides (xy) For each x ЄZ, let [x] be the ~-equivalence class which contains x; and let Z/ ~ be the set of ~-equivalence classes. 1 2 361 PRACTICE FIRST MIDTERM QUESTIONS 2024 (i) Prove that the multiplication operation on Z/ ~ given by is well-defined. x-y=xy (ii) Determine whether the exponentiation operation on Z/ is well-defined. [x][y] = [x] ~ given by

The relation R on the set of all integer numbers is defined by (x, y) = R_if and only if |x| = [y]. Then, R is an equivalence relation. True False

Decide the equivalence classes of the equivalence relation R defined on Z by aRb if 3 | (2a + 7b), how do you determine when to stop? please show the steps to get the equivalence classes in detail, thank you in advance.

The real line R is divided into subsets X1, X2, X3 where X1 = (-00, – 7], X2 = [-7,1), and X3 = [1, 00). Can X1, X2, X3 be equivalence classes with respect to some equivalence relation on R? O No, they can't be equivalence classes for some equivalence relation since X1n X2 + 0. O No, they can't be equivalence classes for some equivalence relation since any equivalence relation on infinite set has infinitely many different equivalence classes. O Yes, these sets can be equivalence classes for some equivalence relation since X1 U X2 U X3 = R. O Yes, these sets can be equivalence classes for some equivalence relation since X2n X3 = 0.

3. (Chapter 11.4, #3) The relation ~ on Z defined by, x ~ y if x = y mod 4, is an equivalence relation. Sketch it, by drawing Z as a subset of the line, then colouring each equivalence class in a different colour.

Let R=}(I,y)|2 divides 8x+by} be a relation the Set H= {1,2,3.4,5,6.10} Determine whether the given relation R is an on equivalence.

What is equivalence relation

We have defined the relations ≡6 and ≡10 to be certain subsets of Z^2 . Their intersection R = (≡6)∩(≡10) is another subset of Z^2 , so it is also a relation on Z. The relation R is equal to ≡m for some natural number m. What is m?

Help please!

Define a relation ~ on P(X), where X = For Y, Z e P(X), we say Y ~ Z if and only if Y has the same number of ele- ments as Z. {1,2, 3, 4, 5, 6}, as follows: List and describe all unique equivalence classes.