## Section Project: More on Inverses, Composition and Relations

###### Problem 2.32.

Is Problem 1.28 valid if we define \((a, b) = \{\{a, 1\}, \{2, b\}\}\text{?}\)

How would one formally define an ordered triple? An ordered quadruple?

###### Problem 2.33.

Show that the composition of one-to-one functions is one-to-one by letting \(g:\B \to \C\) and \(h:\A \to \B\) be one-to-one functions and showing that \(g\circ h:\A \to \C\) is one-to-one.

###### Problem 2.34.

Show that the composition of onto functions is onto.

The word *identity* is used in a number of different contexts in mathematics. The function \(f: \R \to \R\) defined by \(f(x) = x\) is called the *identity* function because it identifies every number with itself. Similarly, \(0\) is called an *additive identity* because adding \(0\) to a number does not change the number, so \(0+x\) is identified with \(x\text{.}\) By the same reasoning, \(1\) is called the multiplicative identity.

###### Problem 2.35.

Let \(\mathbf S\) denote the set of all bijections from \(\A\) to itself. According to Problems 2.33 and Problem 2.34, \(\circ\) is a binary operation on \(\mathbf S\text{,}\) that is, \(f\circ g \in \mathbf S\) whenever \(f\) and \(g\) are in \(\mathbf S\text{.}\) We define a special bijection \(i:\A \to \A\) called the *identity function* on \(\A\) as

Show that \(i\) is an identity for \(\mathbf S\text{,}\) that is,

\(f\circ i = f = i\circ f\) and

\(\displaystyle f\circ f^{-1} = i = f^{-1}\circ f\)

for all \(f \in \mathbf S\text{.}\) This shows that \(i\) serves as an identity for composition in the same way that 0, 1, \(\emptyset\) work as identities for \(+\text{,}\) × and \(\oplus\text{,}\) respectively.

###### Problem 2.36.

Consider the list below of all 6 different bijections from \(\{0,1,2\}\) onto itself. For example \(p\) maps \(0 \to 1\text{,}\) \(1 \to 2\) and \(2 \to 0\text{.}\)

i | p | q | f | g | h | |

012 | 012 | 012 | 012 | 012 | 012 | |

012 | 120 | 201 | 021 | 210 | 102 | |

Make a composition table for these 6 bijections, illustrating Problems 2.33, Problem 2.34 and Problem 2.35 above.

###### Problem 2.38.

A function \(f:\B \to \C\) is said to be *left cancelative* if, for all sets \(\A\) and all functions \(g,h:\A \to \B\text{,}\)

Show that \(f\) is left cancelative if and only if \(f\) is one-to-one.

###### Problem 2.39.

A function \(f:\B \to \C\) is said to be *right cancelative* if, for all sets \(\D\) and all functions \(g,h:\C \to \D\text{,}\)

Show that \(f\) is right cancelative if and only if \(f\) is onto \(\C\text{.}\)