1. Let X be a set with n elements. Compute the number of distinct functions f: X → X. How many of them are injective? How many of them are surjective? How many of them are bijective?

Q: Let A be a non-empty set of real numbers which has a upremum and let B= {x : - x e A}. Show that B…

A:

Q: Let A be a set and let ƒ : A → B be a surjective function. Prove that there exists a subset CCA such…

A: Given: A be a set and f:A→B is a surjective function. To prove: There exists a subset C⊂A such that…

Q: Show that, for any set A, there is exactly one map f from the empty set ∅ to A. When is f injective?…

A:

Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

A:

Q: 3) Let A be a nonempty set of real numbers that is bounded below. Let -A be the set of all numbers…

A:

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A:

Q: 4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then |AN B| 2 3. 5.…

A: We use induction. This clearly holds whenever n=1. Fix n∈N and suppose that every function from…

Q: A relation R is defined on the set of real numbers by xRy if and only if x-y is a multiple of 3.…

A:

Q: 1. Let A be an uncountable set and B C A such that there exists a surjective function f : Z+ → B.…

A:

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

A:

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/xe A}. 1…

A:

Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

A:

Q: Prove whether the following theorem is true or not : If A is any countably infinite set, B is any…

A:

Q: If a is transcendental over F, show that every element of F(a) that isnot in F is transcendental…

A:

Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

A:

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}.…

A:

Q: Suppose (X, S) is a measurable space, E1, . . . , En are disjoint subsets of X, and c1, . . . , cn…

A: Let's suppose that ϕ:X→R defined by, ϕ=c1XE1+c2XE2+⋯+cnXEn So, this function is well defined as…

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

A:

Q: Let R (the set of real numbers) be equipped with the discrete topology, then the semi open interval…

A: Let (X, d) be a discrete metric space. Thus singletons are open sets as {x} = B(x, ϵ) where ϵ <…

Q: 5. Suppose X is a closed set, and f: X R. Prove that f is continuous if and only if, for every…

A: This question is about application of integration and differentiable of functions

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: 48

A:

Q: Prove that for any set A, the identity function iA is a bijection.

A:

Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

A:

Q: Let S be an nonempty open connect subset of R^2. What is the number of possible continuous onto…

A:

Q: A relation R defined on the given set A will satisfy

A:

Q: A relation R defined on the set of real numbers is such that x R yand y Rx >x = y. R is said to be…

A: Properties  used :- (i) a R a  for all a∈ A, [Reflexivity] (ii) a R b  and  b R a ⇒ a = b,…

Q: Let C and D be finite sets and let f :C → D and g : D →C be two functions such that Vy ED f(g(y)) =…

A:

Q: . Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

A:

Q: 2. Consider the relation on the natural numbers N defined as follows: for all x, y e N, xRy if and…

A: According to question,We have to discuss about reflexive, symmetric, transitive relation.

Q: Prove that he interval E = (a, +oo) is an open set in R. (Show all details of your work).

A:

Q: Let A be a non-empty set of positive numbers, which is unded above and let B = {x e R: 1/xe A}. 1…

A:

Q: A relation R, is defined on Z, the set of integers by aRb + a - b is even. Show that R is…

A:

Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A: Remark: Completeness property of R: Every bounded above non-empty subset of R has a least upper…

Q: 4) Let X be a finite set and f : X → X a map.

A: Given, X be a finite set and f : X→X  a map.

Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

A:

Q: Suppose that fis a function from A to B, where A and B are finite sets with |A|=|B|. Then fis…

A: We can solve this using definition of one to one and onto

Q: Let [a,b] be a closed interval which is by a class 9 of open intervals. Prove the contains a finfite…

A:

Q: | Given a finite set A with n elements, let R be the relation defined on the set of functions f:A →…

A:

Q: Let A be a non-empty set and suppose that f : A → Rand g : A → R. Define f + g : A → R by ( ƒ +…

A: we take f(x)=-1   x∈Q∩A1       x∈Qc∩A and g(x)=2   x∈Q∩A-2   x∈Qc∩A(f+g)(x)=f(x)+g(x)=1   x∈Q∩A-1…

Q: Let f : (X,d) - (Y, d1) becontinuous fumction, then for any closed set G in X, f(G) is not necessary…

A: False

Q: Suppose A is an uncountable set and B is a denumerable set. Suppose f:A→B is a function. Prove that…

A:

Q: 2. Let X be an uncountable set and let Y be a countable set. (a) If f : X → Y is a function, prove…

A:

Q: Let there be f: A--&gt;RShow that , if f is bounded in A , then f will also be bounded in B , where…

A: This can be proved as shown below.

Q: 5. (a) How many functions are there from a set with n elements to a set with m elements?

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: Is the set θ={((x,y),(3y,2x,x+y)) : x,y∈R} a function? If so, what is its domain and range? What can…

A:

Q: Let F and G be a non-empty families of sets with SG. Show that ng s UG.

A: Given: F and G are two non-empty family of sets with F⊆G, To show: ∩F⊆∪G