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Chapter 6: Problem 343

Explain why a bijection must have an inverse.

Short Answer

Expert verified
A bijection's unique and complete mappings ensure every element in the codomain can be paired back to an element in the domain, allowing for the construction of an inverse.

Step by step solution

01

Define Bijection

A bijection is a function that is both injective (one-to-one) and surjective (onto). This means every element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped.
02

Bijection Implies Unique Mapping

Because a bijection is one-to-one, each element in the domain maps to a unique element in the codomain, and there are no repeated values in the codomain.
03

Surjectivity Ensures Coverage

Since a bijection is onto, every element in the codomain is associated with some element in the domain. Therefore, there is no element in the codomain that is left unmapped.
04

Constructing the Inverse

Given the properties of bijection, we can construct the inverse by reversing the mapping: for every element in the codomain, assign it back to the unique element in the domain that mapped to it.
05

Verify the Inverse

To verify the inverse, show that composing the bijection with its inverse returns the identity function. Let function f be a bijection from set A to set B, then for any element y in B, let X be the unique element in A such that f(X) = y. The inverse function f鈦宦�(y) will map y back to X.
06

Conclusion

Since a bijection has both unique mapping and complete coverage, it is guaranteed that an inverse function exists, where every element in the codomain maps precisely back to its original element in the domain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bijection
A bijection is a special type of function. It is both injective (one-to-one) and surjective (onto).
This means a bijective function connects every element in the domain (input set) to a unique element in the codomain (output set).
This unique connection ensures there are no duplicates or unmapped elements.

In simpler terms:
  • **Injective:** No two elements in the domain map to the same element in the codomain.
  • **Surjective:** Every element in the codomain has a corresponding element in the domain.
Because of these properties, bijections perfectly pair input and output elements.
Inverse Function
An inverse function reverses the original mapping of a function. If you have a function f that maps elements from set A to set B, the inverse function f鈦宦� maps elements from set B back to set A.

For a function to have an inverse, it must be bijective. That's because bijections ensure that each element in set B has a unique counterpart in set A.
A simple example: if f(x) = y, then f鈦宦�(y) = x.

The essence is:
  • **Reversal:** Inverses undo the action of the original function.
  • **One-to-One Matching:** Only bijections, with their unique and total mapping, can have an inverse.
This is why knowing if a function is bijective helps in determining the presence of an inverse.
Injective and Surjective Mapping
A function being injective and surjective is at the heart of being bijective.
An **injective** function ensures that each element in the domain maps to a unique element in the codomain.
  • Think of it as no two students getting the same grade.
An **surjective** function ensures that every element in the codomain is mapped by some element in the domain.
  • It's like every student gets a grade, and no grade is left out.
When both these conditions are fulfilled, we have a bijection. This means:
  • **Injective:** Prevents overlap in mappings.
  • **Surjective:** Completes the mapping coverage.
Combining both guarantees an inverse function can exist.
Unique Mapping
Unique mapping refers to the one-to-one nature of a bijection.
In a bijective function, every element in the domain maps to a distinct and unique element in the codomain.
This uniqueness ensures there are no repeated values in the codomain and reinforces the injective nature of the function.

Without unique mapping:
  • Two different elements in the domain could map to the same element in the codomain.
  • This would violate the injectiveness and compromise the bijection.
Thus, unique mapping is a cornerstone of bijections and inverses.
Identity Function
An identity function is a function that maps every element to itself.
If you have a set A, the identity function on A, usually denoted by IA, maps each element x in A to x.
So, IA(x) = x for all x in A.
This concept is essential in verifying inverse functions.

When showing a function f and its inverse f鈦宦�, their composition should return the identity function, like:
  • f(f鈦宦�(y)) = y
  • f鈦宦�(f(x)) = x
This demonstrates:
  • **Reversal Property:** The function gets totally reversed back to the original input.
Identity function helps us understand the idea of total and exact reversal in bijections and inverses.

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Most popular questions from this chapter

Another view of the dihedral group of the square is that it is the group of all distance preserving functions, also called isometries, from a square to itself. Notice that an isometry must be a bijection. Any rigid motion of the square preserves the distances between all points of the square. However, it is conceivable that there might be some isometries that do not arise from rigid motions. (We will see some later on in the case of a cube.) Show that there are exactly eight isometires (distance preserving functions) from a square to itself. (h)

In how many ways may we paint the faces of a cube with six different colors, using all six?

Draw the digraph of the relation from the set \\{Sam, Mary, Pat, Ann, Polly, Sarah \(\\}\) to the set \(\\{\mathrm{A}, \mathrm{M}, \mathrm{P}, \mathrm{S}\\}\) given by "has as its first letter."

Suppose we make a necklace by stringing 12 pieces of brightly colored plastic tubing onto a string and fastening the ends of the string together. We have ample supplies blue, green, red, and yellow tubing available. Give a generating function in which the coefficient of \(B^{i} G^{j} R^{k} Y^{h}\) is the number of necklaces we can make with \(i\) blues, \(j\) greens, \(k\) reds, and \(h\) yellows. How many terms would this generating function have if you expanded it in terms of powers of \(B, G, R,\) and \(Y ?\) Does it make sense to do this expansion? How many of these necklaces have 3 blues, 3 greens, 2 reds, and 4 yellows?

Draw the digraph of the relation on the set \\{Sam, Mary, Pat, Ann, Polly, Sarah\\} given by "has the same first letter as."
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