Assume A = R and the relation R ⊆ A × A such that for x,...

Assume A = R and the relation R ⊆ A × A such that for x, y, ∈ R, xRy if and only if sin2 x + cos2 y = 1. Prove that R is an equivalence relation and for any fixed x ∈ R, find the equivalence class x

Expert Solution

Colby Messinger answered 8 months ago

Let x be a set and let R be a relation on x such x is...

Let x be a set and let R be a relation on x such x is simultaneously reflexive, symmetric, and antisymmetric. Prove equivalence relation.

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that...

Let X be a non-empty set and R⊆X × X be an equivalence relation. Prove that X / R is a partition of X.

Consider the equation x^2+(y-2)^2 and the relation “(x, y) R (0, 2)”, where R is read...

Consider the equation x^2+(y-2)^2 and the relation “(x, y) R (0, 2)”, where R is read as “has distance 1 of”. For example, “(0, 3) R (0, 2)”, that is, “(0, 3) has distance 1 of (0, 2)”. This relation can also be read as “(x, y) belongs to the circle of radius 1 with center (0, 2)”. In other words: “(x, y) satisfies this equation if, and only if, (x, y) R (0, 2)”. Does this equation determine a...

6. Let R be a relation on Z x Z such that for all ordered pairs...

6. Let R be a relation on Z x Z such that for all ordered pairs (a, b),(c, d) ∈ Z x Z, (a, b) R (c, d) ⇔ a ≤ c and b|d . Prove that R is a partial order relation.

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...

Suppose x,y ∈ R and assume that x < y. Show that for all z ∈...

Suppose x,y ∈ R and assume that x < y. Show that for all z ∈ (x,y), there exists α ∈ (0,1) so that αx+(1−α)y = z. Now, also prove that a set X ⊆ R is convex if and only if the set X satisfies the property that for all x,y ∈ X, with x < y, for all z ∈ (x,y), z ∈ X.

Consider the following history H: T2:R(Y), T1:R(X), T3:R(Y), T2:R(X), T2:W(Y), T2:Commit, T1:W(X), T1:Commit, T3:R(X), T3:Commit Assume...

Consider the following history H: T2:R(Y), T1:R(X), T3:R(Y), T2:R(X), T2:W(Y), T2:Commit, T1:W(X), T1:Commit, T3:R(X), T3:Commit Assume that each transaction is consistent. Does the final database state satisfy all integrity constraints? Explain.

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if...

Let R be the relation on Z+× Z+ such that (a, b) R (c, d) if and only if ad=bc. (a) Show that R is an equivalence relation. (b) What is the equivalence class of (1,2)? List out at least five elements of the equivalence class. (c) Give an interpretation of the equivalence classes for R. [Here, an interpretation is a description of the equivalence classes that is more meaningful than a mere repetition of the definition of R. Hint:...

Define a relation S from R to R by saying that if and only if (a) List...

Define a relation S from R to R by saying that if and only if (a) List five different elements of S. (b) Prove that S is not a function.

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