Z is the set of integers, R is the set of real numbers. 2) For each of the following functions prove whether they are i) one to one;   ii) onto. If they are not one to one or onto, show why. a) f: Z -> Z, f(n) = n - 6; b) f: Z -> Z, f(n) = n3; c) f: Z -> Z, f(n) = n2 + n;

Q: A relation R defined on the set of real numbers is such that xRy and yRx = x = y: R is said to be A…

A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

Q: Let S be a non-empty open connected subset of R². The number of possible continuous onto functions…

A: Please see the attachment

Q: 1. Show that if f is a bijection from a set X onto a set Y then f is a bijection from Y onto X.

A: Please post the multiple questions separately. Here I answered question (1) only as per our policy.

Q: 2. Let X, Y be nonempty sets of real numbers that are bounded above. Let X + Y = {x + y:x € X and y…

A:

Q: Prove that the set (0. 1) = { all xE RIO <x< 1} is uncountable/non-denumerable.

A:

Q: Consider A defined on R , the set of all real numbers as Ɐ x, y E R , as x A y if and only if xy=1…

A:

Q: 2. Let a, b e R and let f be the function Z + Z defined by f(z) - ar + b. For which values of a and…

A:

Q: Consider two sets, A = {1, 2, 3, ..., 100} and B = {3, 5, 7, 9, ... , 201}. Prove that |A| = |B| by…

A:

Q: Prove that ifS is a bounded set, then inf(S) = -sup(-S) %3D

A:

Q: if f is derivable in [a, b] and f' (a) not equal to f' (b), then for each number k lying between f'…

A:

Q: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

A:

Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

A:

Q: Let (x, eA) be a measurable space, £: X (-∞, w) and 9:* (-∞0, ∞o) are measurable. Prove that the…

A: Given that X,A is a measurable space. The objective is to prove that the sets  {x∈X:f(x)&lt;g(x)},…

Q: s Show that in the metric space (R, 1.1), Q = 4, (Q')"=$ Cinterior of rational numbers and interior…

A: A rational number x is called interior point if there exists a neighbourhood of x which contains…

Q: Consider the collection of open intervals O = 1- In=1 One can show that O is an open cover of the…

A:

Q: Show that any open interval (a,b) of the real numbers has the same cardinality as (0,1). (note that…

A: We have to show that the following statement is true-  Any open interval (a,b) of the real numbers…

Q: Prove that Z ≈ E*, where Z is the set of integers and E* is the set of positive even integers.

A: To prove:

Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers . If x…

A:

Q: ·f is defined as f: R – {3}→ R -, R is the set of real numbers. x+5 f(x) 2х-6 . Is f one-to-one?…

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: Let R be the set of all real numbers with the euclidean topology. Show that R has an uncountable…

A: Let R be the set of all real numbers with the euclidean topology. Show that Rhas an uncountable…

Q: Let (X, CA) be a measurable space, f: X (-00, ws) and 1: X (-∞, ∞0) are measurable. Prove that the…

A:

Q: (a) What is the cardinality of the interval (0, 1), finite, countably infinite, or uncountable? (b)…

A:

Q: snip

A: Inner product of two vectors is equal to sum of product of corresponding elements of the two…

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 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: Prove that the following statement is true for all real numbers x, y ≠ -1, by using a sequence of…

A: y= 1-x/1+x iff x=1-y/1+y proved

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: Let R be an integral domain with unity. State what it means for “being associates” to be transitive,…

A:

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: IS V is finite dimensiona! oad f is a homomor- Ex 6. phisni of V into itself which is not onto prove…

A: Given: V is finite dimensional and f is homomorphism of V into itself such that f is not onto. To…

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: in extension of a fie elements in K which ver F from a subfie

A:

Q: X be a limit point of set ECR, then every o infinitely many points of terval containing

A:

Q: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} issymmetric but neither…

A: Given, relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)}.

Q: Prove disprove that (R.Tco-finite) is Te-space or

A: To Prove - Prove or Disprove that ℝ, τco-finite  is T2 - space.       Definition used - T2 - space…

Q: Show that the class of closed interval [a,b), where a is rational and b is irrationals and a <b, is…

A: Base for topology: A collection B of subsets of topological space X is said to be base for topology…

Q: 20. Give two examples of functions from Z to Z that are one- to-one but not onto.

A:

Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers . If x…

A: To prove: If x&lt;y and y&lt;z then x&lt;z. Proof: The first inequality x&lt;y suggest that the…

Q: Let a and b be constant integers with a * 0, and let the mappingf: Z ----Z be defined by f(x) = ax +…

A: Given:        f:ℤ→ℤ such that,f(x)=ax+b

Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

Q: Suppose A and B are non-empty bounded subsets of positive real numbers. Define A·B = {r •y:x € A, y…

A:

Q: Prove that Yational numbers la b) with equality multiplication and addition defined as Set s of…

A:

Q: 28

A:

Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

A:

Q: 10. Prove that the set H={[,ne subgroap of GL(2, R). For show that x HxcH but xHx+H. :9 GL(2,R),

A:

Q: Find a bijection from the set of positive integers Z+ to the set of integers Z (Thus, conclude Z+ =…

A: the set of positive integers Z+ to the set of integers Z . the function is defined..