A partition P of Z is said to be compatible with + if given any X, Y ∈ P and any x1, x2 ∈ X, y1, y2 ∈ Y the elements x1 + y1 and x2 + y2 belong to the same member of P. Suppose that P is a partition of with 4 members compatible with +. Show that P is the partition associated to the equivalence relation congruence modulo 4. Can you see how to generalize this to a partition of Z compatible with + having an arbitrary amount of classes (not just 4).

Q: With a binary relation R (defined below) on A and each tuple (a, b) indicating that a is related to…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: {k}, {I, m, o}, {n, p}} of the set U = {k, I, m, n, o, p}. Give the

A: Solution is attached below.

Q: 3. Is (A, R) a partial order, a strict order, or neither if A is the set of all the students…

A:

Q: 3.(1) List all the partitions of the set S = {1, 2, 3, 4}. (2) From (1), determine how many…

A: First five partitions are given in step one and the rest is given in step 2  1. 1, 2, 3, 4 2. 1, 2,…

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

A:

Q: Let ∼ ⊆ X ∗X be an equivalence relation on a set X. For x ∈ X define ̄x := {y ∈ X |x ∼ y}. Then it…

A:

Q: the sets  [0,1] and  [0,1/2)∪(1/2,1] have the same cardinality, give an explicit bijection to show…

A: We will define a bijection from the set J  = [ 0 , 12 ) ∪(12 , 1]   to the set I =0 ,1   First we…

Q: 4. Let S = {1, 2,..., 10}. Out of all the equivalence relations on S that have exactly 2 equivalence…

A:

Q: Show that the subset [2,5] [7,9] of R has exactly 4 clopen subset.

A: The question is wrong.

Q: Let Fn be the number of partitions of [n] that dó singleton block. Find a simple relation between…

A:

Q: And what would be the 15 equivalence relations? And what does "up to isomorphism" mean?

A: Ans-In this question we have to determine   all possible equivalence relations on the set…

Q: Calculate the maximum number of binary relations that can be defined on set A = {-1, 0, 1}.

A: We have to find the number of binary relations.

Q: Let X = {a, b, c}. For each of the following binary relations on X, check the boxes next to the…

A:

Q: Let A be the set of letters in the English alphabet with the usual order. Define ⟨A2, ≤⟩ such that…

A:

Q: Consider the equivalence relation ~ on R given by a ~ b if and only if [a] = [b]. You do not need to…

A: Given that the equivalence relation ~ on ℝ defined as a~b if and only if a=b. Define a function…

Q: 5. Consider the relation in N given by n~m if n and m have the same number of digits in the decimal…

A:

Q: 2. Suppose A = {1, 2, 3, 4, .. , 14, 15} and let R be the relation of congruence modulo 5 on the set…

A:

Q: Assume that E, F, G₁, G₂, G., G (wherein i = 1 ton) such that P(EG) is less than or equal to P(FIG)…

A: To show that P(E) ≤ P(F), we can use the law of total probability. We know that the G's form a…

Q: Let σ := (R), where R is a binary relation symbol. Write down a σ-theory T whose models are exactly…

A:

Q: 2. Determine the partition of B = {1,2,3,4,5,6,7,8,9,10} induced by the equivalence relation B…

A:

Q: For an integer n>4, let Sn be the set of all binary strings of length n. The relation R on Sn is…

A: Given,For an integer , let  be the set of all binary strings of length . The relation  on  is…

Q: For a positive integer n ≥ 4, let {0, 1]" denote the set of all binary strings of length n. The…

A: For a positive integer n≥4, let 0,1n denote the set of all binary strings of length n. Observe that,…

Q: 7. A relation R on a set X is called an equivalence relation if it is symmetric, reflexive, and…

A: Equivalence relation: A relation R on a set X is called an equivalence relation if it is symmetric,…

Q: 5. Let A = {41, 49, 67, 71, 73, 76, 81, 88, 111} and R be an equivalence relation defined on A where…

A:

Q: count the number of possible equivalence relations on a set of size n by providing a formula in…

A: We use the idea of partitioning to get the total number of equivalence relations.

Q: 5. Let S be the set of all positive divisors of p?q² where p and q are primes. Consider the poset…

A: Given: Let  Sbe the set of all positive divisors of p2q2 where p and q are primes.…

Q: d) Determine whether Prove it, if not give Ris e) Determine whether If not Ris antisymmetric a…

A:

Q: 30. Let S= {1,2,3} and let R be an equivalence relation on S. (a) Describe the R with the fewest…

A:

Q: 6. Consider the partition P = {{0}, {-1,1}, {-2,2}, {-3,3},{-4,4},...} of Z. Describe the…

A: Given the partition of Z is P=0,-1,1,-2,2,-3,3,-4,4,....

Q: 9. E is the binary relation defined on Z as follows: For all m, n E Z, m Enm-n is even. Is this an…

A: Equivalence relation

Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

A:

Q: Q4. Simplify the following expression: 3-5 x 39 Click on image to enlarge if needed 12 81 243 324

A: To evaluate :- 3-5 * 39

Q: Let A and B be the following sets: A = {m ∈ Z | m = a^2 +3 for some odd integer a}, B = {n ∈ Z | n…

A:

Q: Question No. 5) Show that a partition of a set S determined equivalence Relation on set S. (

A:

Q: Let A = {1,2,3,4,5} B= (6,7,8,9} Define an equivalence relation of p in A UB with atleast 12 members

A: A=1,2,3,4,5B=6,7,8,9A∪B=1,2,3,4,5,6,7,8,9

Q: This is set theory

A: (i) Proving that the multiplication operation on Z/∼ given by [x][y] = [xy] is well-defined:…

Q: Indicate the partition of the set A={ x integer: 2 ≤ x ≤ 12 } resulting from the equivalence…

A:

Q: Theorem 2. Let R and S be binary relations on a set X. Then, (a) (R-1)-1 = R. (b) (RUS)-1 = R-'Us-!.…

A:

Q: Prove the subset relation ⊆ on all subsets of ℤ is a partial order, not a total order. It should…

A:

Q: The approximated value for e using the Maclaurin series expansion coasidering a true relative error…

A:

Q: On S = {3, 5, 7, 8, 9, 10, 12, 27, 30, 70} we work with the divisibility relation R: xRy ⇔ x divides…

A: S = { 3 , 5, 7, 8 , 9 , 10 , 12, 27 , 30 , 70 } The relation on S is x R y ⇋x divides y For a non…

Q: Let A = {55, 63, 70, 83, 86, 106, 113, 116, 151} and R be an equivalence relation defined on A where…

A: Given : A=55, 63, 70, 83, 86, 106, 113, 116, 151 and R is an equivalence relation defined on A where…