3. Show that if X-fryin a connectedcategoryconstant morphism Yis a constant morphionmorphionthen there exists a uniquehY>Y such that hof-f

To show this, let's break it down step by step:1. Constant Morphism: In a category, a morphism f:X→Y…

Answered: 3. Show that if X-ty to a constant…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 54E

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13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
Investigate transfarmation defined as LiR? = B?, L(x,y)=(x+y, x-3) whether the is an isomarphism.
(13) Show that 2Z ∪ 3Z is not a subring of Z.
2. Suppose someone is trying to illustrate Cantor's diagonalization proof that |X # |P(X)|, where X = {a,b, c, d}. Given the hypothetical bijection, f: X P(X), below, what would Cantor's missing element be? x 4 P(X) a {b, a, d} b> {a}
et SCN be the set that contains exactly all prime numbers, and let RC S x S be the relation efined by the rule (x, y) ER⇒ x - y = 2. (a) Prove or find a counterexample for the following properties: • antisymmetry • asymmetry • symmetry ● irreflexivity • reflexivity • transitivity • uniqueness
In this problem you will calculate the area between f(x) = 5x + 9 and the x-axis over the interval [0, 2] using a limit of right- endpoint Riemann sums: Area = lim n→∞ n (2₁ k=1 f(xk) Ax Express the following quantities in terms of n, the number of rectangles in the Riemann sum, and k, the index for the rectangles in the Riemann sum.
Let the domain of x and y be all people. Let W(x): "x is a woman". M(y): "y is a man", and L(x.y): "y loves x". Then vxay. (W(x)AM(y))L(xr, y)] means Every man has a woman that loves him. Every man is loved by all women. None of these Every woman is loved by all men. Every woman has a man that loves her.
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2. Consider morphisms f: A B, g: BC, h: CD, k: DB, j: B → C, E: A → C in a category C so that g is an isomorphism, and all of them satisfy the following equations: koh=g¹, gof=l, ho(jof)-hol. (a) Y *** Draw a digram showing all the functions. (Place the objects A, B, . first and then the morphisms connecting them. You can play around with the positions of the objects, so that. the diagram looks pleasing.) (b) Prove that jofgof. Justify your computations by specifying which property you use at every step. (c) Deduce that if f is an epimorphism, then j = 9.
whenever a E A and Let A and B be non-empty subsets of R with azb bEB. Prove that sup (A) and inf (1B) exist and that sup (A) < Could whenever inf B Sup (A) = inf (B) even though arb AEA and bEB

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3. Show that if X-ty to a constant morphiam in a connected category, then there exist's a unique constant morphism Y. → Y such that hof=f.