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Magazine Chapter 7: Problem 9
Let \(X=\\{1,2,3\\}, Y=\\{1,2,3,4\\}\), and \(Z=\\{1,2\\} .\) a. Define a function \(f: X \rightarrow Y\) that is one-to-one but not onto. b. Define a function \(g: X \rightarrow Z\) that is onto but not one-toone. c. Define a function \(h: X \rightarrow X\) that is neither one-to-one nor onto. d. Define a function \(k: X \rightarrow X\) that is one-to-one and onto but is not the identity function on \(X\).
Short Answer
Expert verified
a. \(f(x) = x + 1\) is one-to-one but not onto.
b. \(g(x) = x \% 2 + 1\) is onto but not one-to-one.
c. \(h(x) = 2\) is neither one-to-one nor onto.
d. \(k(x) = (x \% 3) + 1\) is one-to-one and onto but not the identity function on \(X\).
Step by step solution
01
Define the function
Let us define the function \(f\) as follows:
\[f(x) = x+1\]
for all \(x\in X\).
02
Check one-to-one property
If \(f(x_1) = f(x_2)\) for some \(x_1, x_2 \in X\), then we have
\[x_1 + 1 = x_2 + 1.\]
Subtracting one from both sides, we get
\[x_1 = x_2.\]
So, \(f\) is one-to-one.
03
Check onto property
For the function to be onto, for each element in \(Y\), there must exist an element in \(X\) that maps to it. But we see that there is no element in \(X\) that maps to \(1\) as the minimum value of \(f(x) = x + 1\) is \(2\). Thus, the function is not onto.
b. Define a function \(g: X \rightarrow Z\) that is onto but not one-to-one.
04
Define the function
Let us define the function \(g\) as follows:
\[g(x) = x \% 2 + 1\]
for all \(x\in X\), where \% denotes the modulo operation.
05
Check onto property
Find which elements of \(X\) will map to elements of \(Z\):
\[g(1) = 1 \% 2 + 1 = 2\]
\[g(2) = 2 \% 2 + 1 = 1\]
\[g(3) = 3 \% 2 + 1 = 2\]
As we can see every element in the codomain has at least one element in the domain that maps to it, so the function is onto.
06
Check one-to-one property
We can see that both \(1\) and \(3\) in \(X\) map to \(2\) in \(Z\), i.e., \(g(1) = g(3)\), thus, the function is not one-to-one.
c. Define a function \(h: X \rightarrow X\) that is neither one-to-one nor onto.
07
Define the function
Let us define the function \(h\) as follows:
\[h(x) = 2\]
for all \(x\in X\).
08
Check one-to-one property
Clearly, the function is not one-to-one, as it maps all elements in \(X\) to the same element in \(X\).
09
Check onto property
The function is also not onto, as there is no element in \(X\) that maps to \(1\) and \(3\).
d. Define a function \(k: X \rightarrow X\) that is one-to-one and onto but is not the identity function on \(X\).
10
Define the function
Let us define the function \(k\) as follows:
\[k(x) = (x \% 3) + 1\]
for all \(x\in X\), where \% denotes the modulo operation.
11
Check one-to-one property
If \(k(x_1) = k(x_2)\) for some \(x_1, x_2 \in X\), then we have
\[x_1 \% 3 + 1 = x_2 \% 3 + 1.\]
Subtracting one from both sides, we get
\[x_1 \% 3 = x_2 \% 3.\]
So, \(k\) is one-to-one.
12
Check onto property
Find which elements of \(X\) will map to elements of \(X\):
\[k(1) = 1 \% 3 + 1 = 2\]
\[k(2) = 2 \% 3 + 1 = 3\]
\[k(3) = 3 \% 3 + 1 = 1\]
Every element in the domain maps to a unique element in the codomain, so the function is onto.
13
Check if it's not the identity function
As the function does not map every element in \(X\) to itself, \(k\) is not the identity function on \(X\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Functions in Mathematics
Functions are foundational building blocks in mathematics that establish a relationship between two sets, typically called the domain and the codomain. A function is a rule that assigns each element from the domain to exactly one element in the codomain. This mapping is what makes functions essential in various areas of mathematics and science.
- The domain is the set of all possible inputs for the function.
- The codomain is the set of potential outputs.
- The range is the set of actual outputs that the function can produce.
One-to-one Functions
One-to-one functions, also known as injective functions, are a special type of function where each element in the domain maps to a unique element in the codomain. This means that no two different elements in the domain will map to the same element in the codomain.
- If \( f(a) = f(b) \), then \( a = b \).
- An injective function has distinct outputs for distinct inputs.
Onto Functions
Onto functions, or surjective functions, map every element in the codomain to at least one element in the domain. This means that the range of the function is exactly equal to its codomain.
- For every \( y \) in the codomain, there exists at least one \( x \) in the domain such that \( f(x) = y \).
- All elements in the codomain are "hit" by the function.
Identity Function
An identity function is a simple yet fundamental concept in mathematics. It is a function that maps every element of a set to itself. If the domain and codomain are the same, the identity function will leave each element unchanged as it maps.
- Formally, \( f(x) = x \) for all elements \( x \) in the set.
- Identity functions are both one-to-one and onto.
