Question

Let \(X=\\{-1,0,1,2\\}\) and \(Y=\\{-4,-2,0,2\\},\) Detine the function \(F: X \rightarrow Y\) as \(F(x)=x^{2}-x\). Prove that \(F\) is neither \(1-1\) nor onto.

Short answer

F is neither 1-1 nor onto as multiple inputs give the same output and not all outputs are covered.

Step-by-step solution

Define the Function

The function given is \( F: X \rightarrow Y \) where \( X = \{-1, 0, 1, 2\} \) and \( F(x) = x^2 - x \). Our task will involve evaluating this function for each \( x \) in \( X \) and analyzing the results.

Compute F(x) for each x

Calculate \( F(x) \) for each \( x \) in the set \( X \). We have:- For \( x = -1 \), \( F(-1) = (-1)^2 - (-1) = 1 + 1 = 2 \).- For \( x = 0 \), \( F(0) = 0^2 - 0 = 0 \).- For \( x = 1 \), \( F(1) = 1^2 - 1 = 1 - 1 = 0 \).- For \( x = 2 \), \( F(2) = 2^2 - 2 = 4 - 2 = 2 \).

Analyze the function values for 1-1

To check if \( F \) is 1-1, verify if different elements in \( X \) map to different elements in \( Y \). We observe:- \( F(-1) = 2 \) and \( F(2) = 2 \), thus two different elements in \( X \) correspond to the same element in \( Y \).- Since \( F(x_1) = F(x_2) \) for \( x_1 eq x_2 \) (e.g., \( x_1 = -1 \) and \( x_2 = 2 \)), \( F \) is not 1-1.

Analyze the function values for onto

To determine if \( F \) is onto, check if every element in \( Y \) has a preimage in \( X \).- From our computations, \( F(x) \) obtained the values \( \{0, 2\} \). These are the only values visible in the range.- The elements in \( Y = \{-4, -2, 0, 2\} \) that do not appear as images are \(-4\) and \(-2\) which have no preimage in \( X \).- Since not all elements in \( Y \) are covered, \( F \) is not onto.

Key concepts

1-1 function

In discrete mathematics, a function is considered to be 1-1 (or injective) if it assigns distinct outputs to distinct inputs. This means that if you pick two different elements from the domain, their images must be different as well.
Let's break it down a bit more:

• If you have a function \( f(x) \) and \( f(a) = f(b) \), then it must imply that \( a = b \). In simpler terms, one output for one input.
• If even one pair of different inputs have the same output, then the function is not 1-1.

In our given exercise with the function \( F(x) = x^2 - x \):

• Find that \( F(-1) = F(2) = 2 \).
• Different inputs \( -1 \) and \( 2 \) result in the same output 2.

This observation confirms that the function \( F \) is not 1-1 as different elements in the domain \( X \) have mapped to the same element in the range \( Y \).

onto function

An onto function (also known as a surjective function) covers the entire set of outputs. When a function \( f: X \rightarrow Y \) is onto, every possible element in \( Y \) is attained by some element in \( X \). It means:

• For every \( y \) in \( Y \), there must be at least one \( x \) in \( X \) such that \( f(x) = y \).

In the exercise:

• We found that the range of \( F \) is \( \{0, 2\} \).
• However, the set \( Y \) is \( \{-4, -2, 0, 2\} \).

Since not all elements in \( Y \) (specifically, \(-4\) and \(-2\)) are outputs of the function \( F \), the function is not onto. Essentially, two elements in \( Y \) are left without a pre-image from \( X \).

function mapping

Function mapping is a way to assign each element from one set (the domain) uniquely to an element in another set (the codomain). A function \( F: X \rightarrow Y \) is defined by its rule of association, specifying how elements from set \( X \) map to set \( Y \).

Here's how we typically understand function mapping:

• The function rule tells you what operation to perform to get from one element in the domain \( X \) to an element in the codomain \( Y \).
• The mapping must be consistent, applying the same rule to map every element from \( X \).

In our case, with the function \( F(x) = x^2 - x \), we performed the mapping as follows:

• Elements \(-1, 0, 1,\) and \(2\) from \( X \) were each calculated using \( F(x) \).
• The function value of each results in elements \( 0 \) and \( 2 \) in \( Y \).

The observed mappings illustrate why \( F \) is neither 1-1 nor onto, showcasing the unique results of the function rule within the given domain and codomain.