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# 8 which of the following functions from z to z are one-to-one? Tutorial

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### Expert Maths Tutoring in the UK ^{[1]}

The term one to one relationship actually refers to relationships between any two items in which one can only belong with only one other item. In a mathematical sense, these relationships can be referred to as one to one functions, in which there are equal numbers of items, or one item can only be paired with only one other item

If you are curious about what makes one to one functions special, then this article will help you learn about their properties and appreciate these functions. Using solved examples, let us explore how to identify these functions based on expressions and graphs.

Let’s go ahead and start with the definition and properties of one to one functions.. One to one function is a special function that maps every element of the range to exactly one element of its domain i.e, the outputs never repeat

### Determine whether each of these functions from Z to Z is one-to-one ^{[2]}

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Following our normal procedure to solve this type of exercise, we start with the definitions.. Definition: “A function f is said to be one-to-one, or an injunction, if and only if f (a) = f (b) implies that a = b for all a and b in the domain of f

Remark: “We can express that f is one-to-one using quantifiers as ∀a∀b(f (a) = f (b) → a = b) or equivalently ∀a∀b(a≠b → f (a) ≠f (b)), where the universe of discourse is the domain of the function.” Discrete mathematics and its applications By Rosen.. To prove that the function is one-to-one, we must use the definition

In other words, we will show, using a direct proof, that a-1 = b-1 implies that a = b.. Another useful proof method, studied before, is to find a counterexample.

### Which of the following functions from Z to itself are bijections?A.$f\left( x \right) = {x^3}$B.$f\left( x \right) = x + 2$C.$f\left( x \right) = 2x + 1$D.$f\left( x \right) = {x^2} + x$ ^{[3]}

Which of the following functions from Z to itself are bijections?. Hint: We can find the inverse of each of the given functions

If the function is onto, we can check whether the function is one-one. If a function is both one-one and onto, we can say that it is a bijection.

So, we can check whether each of the given functions is onto and if it is onto, we can check whether it is one-one.. If the function is bijective, the inverse must be defined for all values of the range

### Which of the following functions from Z into Z are bijections? ^{[4]}

Which of the following functions from Z into Z are bijections?. A function is bijective iff it is one-one and onto.

Thus, y = 0 ∈ Z does not have pre image in Z (domain)

### How to show that the following function from $mathbb{Z} to mathbb{Z}$ is injective and/or surjective, or neither? ^{[5]}

Define $$f(n) = frac{n}{2} + frac{1-(-1)^n}{4}$$ for all $n in mathbb{Z}$. Thus, $ fcolon mathbb{Z} to mathbb{Z}$ , where $mathbb{Z}$ is the set of all integers

c) $f$ is a function and is not onto but is one-to-one. d) $f$ is a function and is onto but is not one-to-one

So it is a function, but I am having a hard time showing that it is injective and/or surjective

### 5.3: One-to-One Functions ^{[6]}

We distinguish two special families of functions: one-to-one functions and onto functions. We shall discuss one-to-one functions in this section

Recall that under a function each value in the domain has a unique image in the range. For a one-to-one function, we add the requirement that each image in the range has a unique pre-image in the domain.

A one-to-one function is also called an injection, and we call a function injective if it is one-to-one. A function that is not one-to-one is referred to as many-to-one.

### Which of the Following Functions Form Z to Itself Are Bijections? (A) F ( X ) = X 3 (B) F ( X ) = X + 2 (C) F ( X ) = 2 X + 1 (D) F ( X ) = X 2 + X – Mathematics ^{[7]}

Which of the following functions form Z to itself are bijections?. f is not onto because for y = 3∈Co-domain(Z), there is no value of x∈Domain(Z)

Let y be an element in the co-domain (Z), such that. Let y be an element in the co-domain (Z), such that

[ Rightarrow x = y – 2 in Z left( Domain right)]. [left( c right) fleft( x right) = 2x + 1 text {is not onto because if we take }4 in Zleft( co domain right), then4 = fleft( x right)]

### [Solved] Consider the following statements: 1. A function f : Z  ^{[8]}

A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.. A function f : N → N, defined by f(x) = x + 1, is one-one but not onto.

In words: “Each element in the co-domain of f has a pre-image”. Mathematical Description: f : X →Y is onto ⇔ (rm forall ) y (rm exists )x, f(x) = y

In words: “No element in the co-domain of f has two (or more) pre images” (one-to-one) and “Each element in the co-domain of f has a pre-image” (onto).. A function f : Z → Z, defined by f(x) = x + 1, is one-one as well as onto.

### Sources

- https://www.cuemath.com/algebra/one-to-one-function/#:~:text=A%20function%20that%20is%20not,many%2Dto%2Done%20function.&text=for%20all%20elements%20×1,%E2%89%A0%20g(x2).
- https://computinglearner.com/determine-whether-each-of-these-functions-from-z-to-z-is-one-to-one/
- https://www.vedantu.com/question-answer/which-of-the-following-functions-from-z-to-class-12-maths-cbse-5fa6195457895c0dccb6ee32
- https://philoid.com/question/92884-which-of-the-following-functions-from-z-into-z-are-bijections
- https://math.stackexchange.com/questions/3188262/how-to-show-that-the-following-function-from-mathbbz-to-mathbbz-is-inje
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- https://testbook.com/question-answer/consider-the-following-statements1-a-function–607d327d2971f805de3718e6