7. A relation R on a set X is called an equivalence relation if it is symmetric, reflexive, and transitive. (a) Show that congruence modulo m (the relation on Z where we write x =m y if m|(x − y)) is an equivalence relation. (b) Show that every partition P of a set X defines an equivalence relation declaring ~ y if x and y are in the same partition element. by

Q: Suppose R is a relation defined on the set A = {1,2, 3, 4}. If R is an equivalence relation, what is…

A: Given below the detailed solution

Q: For each set partition P of each set X, describe the induced equivalence relation (a) X = {a, b, c,…

A: Given, For each set partition P of each set X, describe the induced equivalence relation ~p on X.

Q: 6. Let S be a set and suppose is a relation on S. For each of the following, provide a…

A: Let us solve the given problems on relation in the next steps.

Q: In Q1-Q4 prove why each relation has or does not have the properties: reflexive, symmetric,…

A: Since you have asked  multiple questions so as per guidelines we will solve  the first question for…

Q: ~ Define an equivalence relation on Z by a b if a - b is even. Show that . equivalence relation.…

A:

Q: uts) Do the following for the set X = {1,2,3,4,5). 7. (a) Give an example of a relation on X that is…

A:

Q: Define a relation on R by a ~ b iff aª – b² = bª – a². (a) Show that ~ is an equivalence relation.…

A: (a). To show that the given relation is an equivalence relation, we need to show that it is…

Q: 3. Which of the following is an equivalence relation? Explain. (a) lives in the same city as (b) is…

A: Due to Bartleby answering guidelines I have solved only question no 3. Because you didn't mention…

Q: PROVE THAT IF A RELATION X ON NON EMPTY SET B IS REFLEXIVE AND TRANSITIVE THEN THE RELATION X…

A:

Q: Q4) Let S be the set of registration numbers of students enrolled in a BSCS program in a university…

A:

Q: 6. A relation R on a set X is called a total order if it is antisymmetric, reflexive, transi- tive,…

A:

Q: We have defined the relations ≡6 and ≡10 to be certain subsets of Z^2 . Their intersection R =…

A:

Q: ' (a,b) ∼ (c,d) ⇔ a+d =b+c '' defined as '∼' relation is that a equivalence relation?

A: It is equivalence relation if (a, b) belongs to ZxZ

Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: Let a relation ∼ on R2 = {(a, b): a, b ∈ R} be defined as: (a, b) ∼ (c, d) if and only if | a | + |…

A:

Q: Let ~ CX *X be an equivalence relation on a set X. For x EX define x := := {x |x E X} is a partition…

A: This is a question from Relation.

Q: on P(X), where X = Y, Z E P(X), we say Y - Z if and only if Y has the same number of ele- Define a…

A: Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say  Y~Z if and only…

Q: 3. Show that the relation R (called lexicographic ordering) defined on Z x Z by: (a, b)R(c, d) if (a…

A: The lexicographic ordering relation R on ℤ×ℤ is defined by, a, bRc, d if a>c or a=c and b≥d Show…

Q: A relation ''<'' is 1) İrreflexive. 2) Transitive. called a quasi-order on a set S if its: صواب ihi

A: We know that

Q: 9. Define a relation R on {-2,-1,0, 1, 2, 3, 4, 5, 6, 7} by a R b if and only if 3a+b is a multiple…

A:

Q: Show that the intersection of any collection of equivalence relations over a set A is also an…

A: We have to prove intersection of equivalence relations on a set is equivalence Let us suppose R and…

Q: Decide whether each of the following relations is an equivalence relation or not.  {(a, b) | gpa of…

A: To prove equivalence relation, we have to prove that (i) the relation is reflexive, (ii) the…

Q: Show that the following is an equivalence relation: R = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3,…

A: Equivalence relations are a fundamental concept in mathematics that allow us to group elements of a…

Q: (i) The equivalence class [b] determined by b E B is a non-empty set. 1 (ii) If r and y are elements…

A: Given that S and T are equivalence relation on a set B. i ) the equivalence class [b] determined by…

Q: Let A be a set and R = {(x, x) : x ∈ A} be the is-equal-to relation on A. Prove that a relation S on…

A: We are given any non-empty set A, on which a relation R = { (x, x) : x∈A } is defined.  Let S be any…

Q: In each of the following cases of a set X and relation determine whether ~ is reflexive, symmetric,…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: (a) Show that R is an equivalence relation. (b) List three members of each of the equivalence…

A: Equivalence relation: A relation R is called an equivalence relation in a set A, if it follows the…

Q: Exercise 1.2. Determine whether the following relations ~ is an equivalence relation: (1) nm in Z if…

A: Step 1:Step 2:Step 3: Step 4:

Q: 1. Suppose A = {1,2, 3, 4} and the relation R on A is R = {(1,1), (2, 2), (3,3), (3, 4), (4, 4)} (a)…

A:

Q: Give an example of an equivalence relation R on the set Q such that there are exactly three distinct…

A: We take the help of the signum function

Q: Suppose 1 and ₂ are equivalence relations on a set A. Let ~ be defined by the condition that a~ b if…

A:

Q: Which of the following is an equevalence relation on the set (a, b, c} 7 O (a, a), (a, b), (b, a).…

A: Option 1 is equivalence relation. Option 2 is also is an equivalence relation. Option 3 and 4 is not…

Q: (Chapter 11.4, #3) The relation relation. Sketch it, by drawing Z as a subset of the line, then…

A: The objective is to show that x≡ymod4 is an equivalence relation. To show the relation is an…

Q: 5. Let X be the set X = {1,2,3}, and let R be the relation defined on X by x₁ Rx₂ if x1 + x2 3. (a)…

A:

Q: The relation R=((a,a),(b,b)) on set X=(a,b) is. transitive symmetric reflexive anti-symmetric None…

A:

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

A:

Q: To find an equivalence class of x ϵ X determined by the relation R ⊆ X ⨯ X, R must be an equivalence…

A: Equivalence Relation: A relation R in X is called an equivalence relation if it is reflexive,…

Q: Let A # 0 be a set. Prove that A? is an equivalence relation on A.

A:

Q: (a) Show that the relation defined on Z by m~n if 4m +n is divisible by 5 2 is an equivalence…

A:

Q: A relation on a set A is circular iff , y, z x~y and y~z implies z~x. Prove that a relation is an…

A: The relation on a set is a subset of the Cartesian product of the set. Therefore, each element is in…

Q: 7. Let A = {1,2,3,4}x{1,2,3,4}. Define an equivalence relation ~ by (x1,x2) ~ (x3,xa) iff xxx2 =…

A: Let A=1,2,3,4×1,2,3,4. The equivalence relation ~  is defined by,        "x1,x2~x3,x4 only if…