Problem 3 - Applied Discrete Structures

Question

Which of the following are one-to-one, onto, or both? (a) \(f_{1}: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f_{1}(x)=x^{3}-x .\) (b) \(f_{2}: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f_{2}(x)=-x+2\). (c) \(f_{3}: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f_{3}(j, k)=2^{j} 3^{k}\) (d) \(f_{4}: \mathbb{P} \rightarrow \mathbb{P}\) defined by \(f_{4}(n)=\lceil n / 2],\) where \([x]\) is the ceiling of \(x\), the smallest integer greater than or equal to \(x\). (e) \(f_{5}: \mathbb{N} \rightarrow \mathrm{N}\) defined by \(f_{5}(n)=n^{2}+n\) (f) \(f_{6}: \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}\) defined by \(f_{6}(n)=(2 n, 2 n+1)\).

Step-by-Step Solution

Answer

f1 is onto, f2 is bijective, f3 and f6 are one-to-one, f4 and f5 are neither.

Step 1: Analyze Function f1

Function \(f_{1}(x) = x^{3} - x\) is a polynomial function mapping from \(\mathbb{R}\) to \(\mathbb{R}\). To check if it is one-to-one, we solve \(f_1(x_1) = f_1(x_2)\) leading to \(x_1^3 - x_1 = x_2^3 - x_2\). Simplifying, \( (x_1-x_2)(x_1^2+x_1 x_2+x_2^2-1)=0\). This equation shows that different \(x\) values can produce the same \(f_1(x)\), so the function is not one-to-one. For onto, \(f_1:\mathbb{R}\to \mathbb{R}\) covers all real numbers because polynomials are surjective over the reals. Therefore, \(f_1\) is only onto.

Step 2: Analyze Function f2

Function \(f_{2}(x) = -x + 2\) is a linear function from \(\mathbb{Z}\) to \(\mathbb{Z}\). It is one-to-one because if \(f_{2}(x_1) = f_{2}(x_2)\), then \(-x_1 + 2 = -x_2 + 2\) which implies \(x_1 = x_2\). It is onto because for any \(y \in \mathbb{Z}\), choose \(x = -y + 2\) which maps back to \(y\). Therefore, \(f_2\) is both one-to-one and onto.

Step 3: Analyze Function f3

Function \(f_{3}(j, k) = 2^{j}3^{k}\) is mapping natural number pairs to \(\mathbb{N}\). To be one-to-one, \(f_3(j_1, k_1) = f_3(j_2, k_2)\) implies \(2^{j_1}3^{k_1} = 2^{j_2}3^{k_2}\), leading to \(j_1 = j_2\) and \(k_1 = k_2\). So, \(f_3\) is one-to-one. It is not onto because not all natural numbers can be expressed as \(2^j 3^k\), such as the prime number \(5\). Thus, \(f_3\) is one-to-one but not onto.

Step 4: Analyze Function f4

Function \(f_{4}(n) = \lceil n/2 \rceil\) maps primes to primes. It is not one-to-one because if \(n\) is even and \(n+1\) is odd, both map to the same ceiling value. For example, \(\lceil 2/2 \rceil = \lceil 3/2 \rceil\). It is neither onto because not all primes can be generated (e.g., \(5\)). Hence, \(f_4\) is neither one-to-one nor onto.

Step 5: Analyze Function f5

Function \(f_{5}(n) = n^2 + n\) maps natural numbers to natural numbers. This is not one-to-one because different inputs can provide the same output, e.g., \(f_5(0) = 0\) and \(f_5(-1) = 0\). It is not onto as numbers like 7 cannot be represented by \(n^2+n\). Therefore, \(f_5\) is neither one-to-one nor onto.

Step 6: Analyze Function f6

Function \(f_{6}(n) = (2n, 2n+1)\) maps natural numbers to pairs of even and odd numbers. It is one-to-one because distinct \(n\) gives distinct pairs. It is not onto since not all pairs \((a, b)\) can be generated (e.g., \((1, 2)\) is not viable). Therefore, \(f_6\) is one-to-one but not onto.

Key Concepts

One-to-One Function

A one-to-one function, also known as an injective function, matches each element from the domain to a unique element in the codomain. In simpler terms, no two different inputs produce the same output. For example, if we have function \(f(x)\) such that \(f(a) = f(b)\) implies \(a = b\), then \(f\) is one-to-one.

• Ensures unique mapping for each input.
• No duplicates in function outputs for distinct inputs.
• Function \(f_2(x) = -x + 2\) from the exercise is a classic example of a one-to-one function. The equation \(-x_1 + 2 = -x_2 + 2\) implies that \(x_1 = x_2\), meeting our one-to-one criteria.

Recognizing whether a function is one-to-one involves setting the outputs of two inputs equal and solving to check if the only solution is when the inputs themselves are equal.

Onto Function

An onto function, or surjective function, covers every element in the codomain with the image of an element from the domain. In other words, every possible output value can be obtained by applying the function to some input value. This means your function's range is equal to its codomain.

• Surjective when every element in the codomain is the function value of some element in the domain.
• Examples: Function \(f_1(x) = x^3 - x \) is onto when mapping from \(\mathbb{R}\) to \(\mathbb{R}\), since for every real number \(y\), you can find an \(x\) such that \(f_1(x) = y\).

Checking if a function is onto involves showing that for every potential output, there exists an input corresponding to it.

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents with constant coefficients. These are among the most basic of all functions and are often denoted as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + ax + c\).

• Includes terms like \(x^3 - x\) as seen in \(f_1(x)\).
• Polynomial functions are continuous and differentiable.
• They can be onto over \(\mathbb{R}\) as indicated in the exercise for \(f_1\).

Understanding polynomial functions can help in analyzing their behavior over different domains and defining characteristics for further mathematical operations.

Natural Numbers

Natural numbers are the set of positive integers beginning from 1 onwards. This set is sometimes represented as \(\mathbb{N}\) and includes numbers like 1, 2, 3, and so on.

• Used in mathematical functions that involve counting or ordering.
• Many mathematical functions, such as \(f_3(j, k) = 2^j 3^k\), utilize natural numbers for inputs and outputs.

Understanding the properties of natural numbers is essential when determining domains and ranges for functions that map from or to this set.

Mapping in Mathematics

Mapping, in mathematics, refers to the process of assigning each item in one set to an item in another set. It's often depicted as a function and defined by statements such as \(f: A \rightarrow B\), where every element of set \(A\) is assigned to an element in set \(B\).

• In essence, a method to relate two sets of objects, numbers, or elements.
• This concept underlies the discussion of functions like \(f_2(x) = -x + 2\) which maps integers to integers, demonstrating both injective and surjective properties.
• Mapping can involve direct, linear, or complex mathematical relationships.

Comprehending mapping is crucial for understanding how mathematical operations transform data and how that data can then be related or connected through functions.