**Title: Functions and Their Properties****Introduction:**In mathematics, understanding the properties of functions is essential. Here, we explore the conditions under which a map between two sets, \(X\) and \(Y\), is injective or surjective.**Content:****Let \( f: X \to Y \) be a map between the sets \(X\) and \(Y\).****(a) Injectivity:**- *Definition:* A function \(f\) is injective (or one-to-one) if distinct elements in the domain map to distinct elements in the codomain. In other words, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\).- *Statement:* Show that \( f \) is injective if and only if there exists a map \( g: Y \to X \) such that \( g \circ f = \text{id}_X \). - *Explanation:* Here, \(\text{id}_X\) denotes the identity map on \(X\), which maps every element of \(X\) to itself. The condition \( g \circ f = \text{id}_X \) means that applying \( f \) followed by \( g \) brings any element back to its original value in \(X\).**(b) Surjectivity:**- *Definition:* A function \(f\) is surjective (or onto) if every element in the codomain \(Y\) is mapped to by at least one element in the domain \(X\).- *Statement:* Show that \( f \) is surjective if and only if there exists a map \( h: Y \to X \) such that \( f \circ h = \text{id}_Y \).- *Explanation:* Here, \(\text{id}_Y\) denotes the identity map on \(Y\). The condition \( f \circ h = \text{id}_Y \) ensures that every element in \(Y\) is the image of some element from \(X\).**Conclusion:**Understanding these properties helps in analyzing the behavior of functions and their inverses. The existence of mappings \(g\) and \(h\) illustrating these properties is a cornerstone in mathematical analysis and set theory.

(a). Let’s first assume that F is an injective map.We need to show that there exists a map such…

Let f: X→ Y be a map between the sets X and Y. (a) Show that f is injective if and only if there exists a map g: Y → X so that go f = idx. (b) Show that f is surjective if and only if there exists a map h: Y → X so that foh = idy.

Let f: X→ Y be a map between the sets X and Y. (a) Show that f is injective if and only if there exists a map g: Y → X so that go f = idx. (b) Show that f is surjective if and only if there exists a map h: Y → X so that foh = idy.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let A = {x, y, z}. Write down a relation R on A that is not reflexive, not sym- metric, and not…

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

Q: Let X and Y be topological spaces and fix some yo E Y. Show that the map f: X→XXY defined by f(x) =…

Q: (b) Let A be a partition of X. Prove that R = {(x, y) = X : there exists i such that x € A; and y €…

A: A is a partition of X.R=x, y∈X: there exists i such that x∈Ai and y∈Ai

Q: Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then…

Q: Let f: A --> B and g: B --> C be maps. Show that if g of f is 1-1 and onto, then f is 1-1 and g is…

A: According to the given information,

Q: (29) Let~ be the relation on R given by y Prove that is an equivalence relation on R. Find [].…

A: A relation ~ on the set of real numbers R is defined as x~y if and only if x-y∈Z. Our aim is to show…

Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, B, 7, 8}, and ƒ the map f(1) = a f(2)= a ƒ(3) = d…

Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

Q: Let X be the set {1, 2, 3, 4, 5, 6}, Y the set {a, ß, 7, 6}, and ƒ the map f(1) = a f(2)= a f(3) = a…

Q: (a) What is the image of [1, 5]? (b) What is the image of [-3, 2]?

Q: C: Prove that the linear map T(x1, x2, x3, x4) = (2x2, 3x3, 4x4, x1) is an isomorphism of R4 to…

Q: 2. Prove that ƒ : (R × R, +) → (R, +) given by f(x, y) = x + y is a homomorphism. Is g : (R × R, +)…

Q: 2. Let : RR and y: R+ → R be mappings with 6(x) -2e-x. (a) Find the range of y. = 1. Use your answer…

A: Here's a summary of the answers: Solutions:(a) Find the range of γ (b) Use the definition to show…

Q: Let D= {1, 2, 3, 4). Give an example of a relation W on D such that W is anti- symmetric but not…

Q: Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3,…

Q: Let T : R3 → R² be a linear mapping. 2 .T Given that T and T find T

Q: Determine if the map g : Q ! Q dened by g(x) = x/(x2+1) is one-to-one.

Q: If n<m then every linear map T: IRn -> IRm is one-to-one. Is this statement true or false?

A: The given statement is

Q: (b) Let {A₁, A2,...} be a partition of a set X, and R the relation {(x, y) = X²: there exists j such…

A: An equivalence relation is a binary relation between two elements of a set that satisfies three…

Q: Suppose q: X → Y is a surjective map. Show that q is a quotient map if and only if the following…

Q: Suppose T: V --> W is a linear and surjective map and V and W are of finite dimension. Show that T…

Q: Let R be a relation from X to Y and S be its reverse relation from Y to X. Prove or disprove that R…

A: According to the given information it is required to prove/disprove that R is injective and well…

Q: Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise…

Question

**Title: Functions and Their Properties**

**Introduction:**
In mathematics, understanding the properties of functions is essential. Here, we explore the conditions under which a map between two sets, \(X\) and \(Y\), is injective or surjective.

**Content:**

**Let \( f: X \to Y \) be a map between the sets \(X\) and \(Y\).**

**(a) Injectivity:**
- *Definition:* A function \(f\) is injective (or one-to-one) if distinct elements in the domain map to distinct elements in the codomain. In other words, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\).
- *Statement:* Show that \( f \) is injective if and only if there exists a map \( g: Y \to X \) such that \( g \circ f = \text{id}_X \).
- *Explanation:* Here, \(\text{id}_X\) denotes the identity map on \(X\), which maps every element of \(X\) to itself. The condition \( g \circ f = \text{id}_X \) means that applying \( f \) followed by \( g \) brings any element back to its original value in \(X\).

**(b) Surjectivity:**
- *Definition:* A function \(f\) is surjective (or onto) if every element in the codomain \(Y\) is mapped to by at least one element in the domain \(X\).
- *Statement:* Show that \( f \) is surjective if and only if there exists a map \( h: Y \to X \) such that \( f \circ h = \text{id}_Y \).
- *Explanation:* Here, \(\text{id}_Y\) denotes the identity map on \(Y\). The condition \( f \circ h = \text{id}_Y \) ensures that every element in \(Y\) is the image of some element from \(X\).

**Conclusion:**
Understanding these properties helps in analyzing the behavior of functions and their inverses. The existence of mappings \(g\) and \(h\) illustrating these properties is a cornerstone in mathematical analysis and set theory.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: With the help of injective function property proof part (A).

VIEW

Step 2: Detailed proof part (b).

VIEW

Solution

VIEW

Step by step

Solved in 3 steps with 18 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Prove that if f: R→ R is continuous, then the image of a bounded set over f is bounded.
how would you prove this
Let T : V → W be a linear map. Show that T preserves linear combinations, i.e. T(a1V1 + ... + anVn) = a1T(v1) + .... + anT(Vn).
Let us define a relation S on R3: If p1 = (x1, y1, z1) and p2 = (x2, y2, z2) are two points in R3, then (p1, p2) in S (or p1 ~ p2) if (x1 - x2) + 2(y1 - y2) + 3(z1 - z2) = 0. Describe all equivalence classes. [Continuation of an earlier question.]