Suppose we define a relation on the set of natural numbers as follows. Two numbers are...

Suppose we define a relation on the set of natural numbers as follows. Two numbers are related iff they leave the same remainder when divided by 5. Is it an equivalence relation? If yes, prove it and write the equivalence classes. If no, give formal justification.

Expert Solution

Yes,it is an equivalence relation.

Colby Messinger answered 2 years ago

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0}...

Suppose we define a relation ~ on the set of nonzero real numbers R* = R\{0} by for all a , b E R*, a ~ b if and only if ab>0. Prove that ~ is an equivalence relation. Find the equivalence class [8]. How many distinct equivalence classes are there?

Suppose we define the relation R on the set of all people by the rule "a...

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Given a natural number q ≥ 1, define a relation ∼ on the set Z by...

Given a natural number q ≥ 1, define a relation ∼ on the set Z by x ∼ y if x - y is divisible by q. (i) Show that ∼ is an equivalence relation. We will denote the set of equivalence classes defined by ∼ with Z=qZ. Also let x mod q denote the equivalence class to which an integer x belongs. (ii) Check that the operations (x (x mod q) + (y mod q) · (y mod q)...

Define d to be the set of all pairs (x,y) of natural numbers such that x...

Define d to be the set of all pairs (x,y) of natural numbers such that x divides y. Show that N is partially ordered by d. Define d analogously on Z. Is then d also a partial order on Z?

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff...

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements. 3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is...

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

Suppose A is the set of positive real numbers, and suppose u and v are two...

Suppose A is the set of positive real numbers, and suppose u and v are two strictly increasing functions.1 It is intuitive that u and v are ordinally equivalent, since both rank larger numbers higher, and therefore generate the same ranking of numbers. Write this intuition as a proof.

Let p and q be any two distinct prime numbers and define the relation a R...

Let p and q be any two distinct prime numbers and define the relation a R b on integers a,b by: a R b iff b-a is divisible by both p and q. For this relation R: Prove that R is an equivalence relation. you may use the following lemma: If p is prime and p|mn, then p|m or p|n. Indicate in your proof the step(s) for which you invoke this lemma.

2. Let D be a relation on the natural numbers N deﬁned by D = {(m,n)...

2. Let D be a relation on the natural numbers N deﬁned by D = {(m,n) : m | n} (i.e., D(m,n) is true when n is divisible by m. For this problem, you’ll be proving that D is a partial order. This means that you’ll need to prove that it is reﬂexive, anti-symmetric, and transitive. (a) Prove that D is reﬂexive. (Yes, you already did this problem on one of the minihomework assignments. You don’t have to redo the...

Let A be a set of real numbers. We say that A is an open set...

Let A be a set of real numbers. We say that A is an open set if for every x0 ∈ A there is some δ > 0 (which might depend on x0) such that (x0 − δ, x0 + δ) ⊆ A. Show that a set B of real numbers is closed if and only if B is the complement of some open set A

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