1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a counterexample.

Q: (c) Show that if g o f is onto then g must be onto.

A: Solution:-

Q: Convert the following formula into one that has neither an existential quantifier nor a Skolem…

A: convert the following formula into one that has neither an existential quantifier nor a skolem…

Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊mn ⌋.

A:

Q: Prove that if a, b e Z – {0}, then gcd(a, b) = gcd(a, -b)

A: Solution:-   gcd a,b= Greatest common divisor of  a  and  b ∗Every integer  that divides  a  and  b…

Q: Let 2= {p₁₁₁r, s, t, u,v} and d = {{p.a.v}, {st.v}-{4.0}} q, U, V 1 1 field containing d. construct…

A:

Q: Prove by bijective (one to one onto) of function 2" f: 2ln] P(({0, 1}, n)).

A: In this question we have to deal with the cardinality of a function.

Q: Let f : R → R be f(r) = + 1+ \|° (a) Is ƒ everywhere defined? If not, give the domaim (b) Is ƒ onto?…

A: Given, f: R→R be                       f(x)=x1+|x|

Q: C. How many extensions of g to A are onto?

A:

Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: It follows by definitic ecause F is onto, there exists an x in X, such that F(x) = y image of a set…

A: Everything You have done was correct except the second last step . So, I m doing from there . Since…

Q: Let S and T be subset of R. Prove the following: cl(SUT) = (cl S) U (cl T)

A:

Q: Define T R → R by T (a, b, c) = (a – c, b+ c) Is T one to one? [ Type Y for yes and N for no. Is T…

A: T  can't be one-one because of domain and codomain but T is onto

Q: Exercise 8.4.19. For each of the following functions, either prove that it is onto, or prove that it…

A:

Q: Show that given relation is Symmetric For X, y e R say x is congruent to y modulo Z if x - y is an…

A: This question is related to number theory.

Q: If E, and E2 are measurable then (E,UE,) is not measurable .

A:

Q: Let =[2,0}, then f is o- algebra.

A: Given a statement

Q: Suppose that f: R20 R20 is defined by f(x) = x². What can we say about f a.f is one to one but it is…

A:

Q: f:A→Z is not onto and f:A M is onto.

A:

Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

A:

Q: Let A be a finite set, and suppose that f : A → A is one-to-one. Does f necessarily have to be onto?…

A: The given question is solved below.

Q: If E, and E2 are measurable , then (E,UE,) is not measurable . صواب İhi

A: Here we have given E1 and E2 are measurable sets

Q: (b) If P is the projection on ce M of a Hilbert space H th riant under an operator T

A:

Q: Are f and g both necessarily onto, if gof is onto?

A: Given: f and g are two functions and gof is onto.

Q: Give an explicit formula for a function from Z to Z+ that is a) onto, but not one-to-one. b)…

A:

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

A:

Q: Let S be the set of integers. If a, b E S, define aRb if ab &gt;= 0. Is R anequivalence relation on…

A:

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: To be onto, for any integer y, there exists an integer x such that y = f(x). Give a counterexample,…

A: To be onto for any real number y, there exists a real number x such that y = f(x). Let y be a real…

Q: Let h: (Z, +) (R – {0},.) define as h(n) = n² then h is 1-1 O 1-1 and onto onto No one O O O

A: If a horizontal line can intersect the graph of the function only a single time, then the function…

Q: (c) Suppose n = 5 and m = 4, determine the number of onto functions from A to B.

A:

Q: Let x, y ∈ Z. Prove that x − y is even if and only x and y are of the same parity.

A:

Q: 6. Is the function h: Z → Z defined by ( 2n if n 20 h(n) -n if n < 0 one-to-one? Is it onto?

A: The solution is given as follows

Q: Suppose f : N→N and g :N→N , function f is not onto but f+g is onto function  does g onto function…

A: g is not necessary to be onto. See the attachment

Q: Prove that G and H IS sube GroP form CX H

A:

Q: Is the function f(n) = n - 5 from Z to Z (where Z is the set of integers) (a) one-to-one? Explain…

A:

Q: i. Prove that gi : Z → Z; g1 (x) = x – 4 is both one-to-one and onto. ii. Prove that g2 : R → R;…

A:

Q: 20. If f: X → Y and g: Y → Z are functions and gof is onto, must ƒ be onto? counterexample.

A:

Q: Determine whether the function f: Z x Z -→ Z is onto if f (m, n) = .m + n

A: We will solve the problem using the definition of onto function.

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

A:

Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

A:

Q: If V is finite dimensional ond f is a homomorphisni of V into itself which is not onto prove that…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Let fons =loag, Cln). Find t'ot at

A: Given, f(x)=logx(ln(x))

Q: Suppose functions w: PQ and h : Q+ R are both onto. Prove that the composite function is also onto.

A:

Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: In Z, let H = &lt;5&gt; and K = &lt;7&gt;. Prove that Z = HK. Does Z = H X K?

A: Given: In Z H=5K=7

Q: If ToS: V Z is onto, then T is onto.

A:

Q: Let Rt st a a, b€ Z and let D=R a-4. 9. show that o E homomosphic Romemerphic Momor

A:

Q: Suppose f : N → N and g : N → N are functions such that f is not onto but f + g is onto. May we…

A: Given: Two functions f : N → N and g : N → N are functions such that f is not onto but f + g is…