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.

Expert Solution

Colby Messinger answered 2 years ago

Is the relation R consisting of all ordered pairs (a, b) such that a and b...

Is the relation R consisting of all ordered pairs (a, b) such that a and b are people and have one common parent: reflexive, irreflexive, symmetric, antisymmetric, asymmetric, and/or transitive? If a property doesn’t hold give a counter-example and state the logical definitions of the properties as you consider them.

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:...

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 S = {2 k : k ∈ Z}. Let R be a relation defined on...

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

let R = Z x Z. P be the prime ideal {0} x Z and S...

let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and...

Let S be the set of all ordered pairs of real numbers. Deﬁne scalar multiplication and addition on S by: α(x1,x2)=(αx1,αx2) (x1,x2)⊕(y1,y2)=(x1 +y1,0) We use the symbol⊕to denote the addition operation for this system in order to avoid confusion with the usual addition x+y of row vectors. Show that S, together with the ordinary scalar multiplication and the addition operation⊕, is not a vector space. Test ALL of the eight axioms and report which axioms fail to hold.

Let V be the set of all ordered pairs of real numbers. Consider the following addition...

Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations V. Let u = (u1, u2) and v = (v1, v2). Show that V is not a vector space. • u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 ) • ku = (ku1 + k − 1, ku2 + k − 1) 1)Show that the zero vector is 0 = (−1, −1). 2)Find the...

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.

6. (a) let f : R → R be a function defined by f(x) = x...

6. (a) let f : R → R be a function defined by f(x) = x + 4 if x ≤ 1 ax + b if 1 < x ≤ 3 3x x 8 if x > 3 Find the values of a and b that makes f(x) continuous on R. [10 marks] (b) Find the derivative of f(x) = tann 1 1 ∞x 1 + x . [15 marks] (c) Find f 0 (x) using logarithmic differentiation, where f(x)...

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