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\[\{1,2,3,4,5\}\setminus\{2,5,6,7\}=\{1,3,4\}.\]
* **Complement.** If \(B\subset A\), then the setwise difference \(A\setminus B\) is called the _complement_ of \(B\) in \(A\). For example, the complement of \(\{1,2,3\}\) in \(\{1,2,3,4,5\}\) is \(\{4,5\}\). In particular, if we are working within a universal set \(X\), then the complement of a set \(S\subset X\) is the set of all things outside of \(S\).
* for example, the set \(X\times X\times X=X^{3}\) is the set of ordered triples of elements of \(X\), and \(X^{n}\) is the set of ordered \(n\)-tuples (i.e., ordered lists of length \(n\)) of elements of \(X\). For example, the set \(\mathbb{R}\) of real numbers forms a line, while the set \(\mathbb{R}\times\mathbb{R}=\mathbb{R}^{2}\) forms a plane, etc.
Set theory is at the core of everything we do in mathematics. For instance, multiplication arises from the Cartesian product – in the example above, the product of a two-element set with a three-element set is a \(2\times 3=6\) element set. In the next section we will see how logic itself arises from set theory. For the remainder of this section, we will see one of the most important ideas in all of mathematics, the idea of a _function_.
### Functions
Let \(X\) and \(Y\) be sets. A _function_ or _mapping_\(f:X\to Y\) from \(X\) to \(Y\) is a rule \(f\) which takes input values \(x\) from \(X\) and assigns to each \(x\) a _unique_ output value \(f(x)\in Y\), as seen in Figure 1.4.
A function can be conceptualized as machine turning elements of \(X\) into elements of \(Y\). Formally speaking, a function \(f\) is a subset of the Cartesian product \(X\times Y\) where each element \(x\in X\) appears exactly once in the first position so that each \(x\) gets an unambiguously defined \(f(x)\). For example, let \(X=\{1,2,3\}\) and \(Y=\{a,b,c,d\}\). Then the Cartesian product is
\[X \times Y\] \[=\{(1,a),(1,b),(1,c),(1,d),(2,a),(2,b),(2,c),(2,d),(3,a),(3,b),(3,c ),(3,d)\}.\]
On the other hand, there is a function \(f:X\to Y\) defined by \(f(1)=b\), \(f(2)=d\), and \(f(3)=d\); we can identify this set with the subset
\[f=\{(1,b),(2,d),(3,d)\}\subset X\times Y. \tag{1.1} \]\]
Almost everything in mathematics involves functions. Calculus is largely the study of functions \(f:\mathbb{R}\to\mathbb{R}\), where \(\mathbb{R}\) is the set of _real numbers_; multivariable calculus involves the study of functions of the form \(f:\mathbb{R}^{n}\to\mathbb{R}^{m}\). Linear algebra, abstract algebra, topology, geometry, and analysis all come down more or less to the study of functions satisfying certain properties.
One way to define a function is with a formula, e.g., \(f:\mathbb{R}\to\mathbb{R}\) is given by
\[f(x)=x^{3}-\sin x\quad\text{or}\quad f(x)=\left\{\begin{array}{ll}x^{2}-1&:x \leq 0\\ \sqrt{x+1}&:x>0.\end{array}\right.\]
A function \(f:\mathbb{R}\to\mathbb{R}\) can be defined with a graph, i.e., a curve that does not intersect any vertical line more than once. Then each point has coordinates \((x,f(x))\). From the graph we can see very clearly that the function is a subset of the cartesian product space \(\mathbb{R}\times\mathbb{R}\), i.e., the \(xy\)-plane.
Let \(f:X\to Y\) be a function. The set of input values \(X\) is called the _domain_ of the function \(f\), and the set \(Y\) of possible output values is called the _codomain_ of \(f\). The _image_ of \(f\) is the set
\[\text{Im}(f)=f(X)=\{y\in Y\ |\ y=f(x)\text{ for some }x\in X\}.\]
Figure 1.4: A function \(f\)The image of a function is always a subset, not necessarily proper, of the codomain. Note that some authors use the term “range” to refer to the image of \(f\), while others use the term “range” to mean the codomain of \(f\). We will prefer to avoid using the term “range” altogether to avoid confusion.
For any subset \(S\subset Y\), the _preimage_ of \(S\) is
\[f^{-1}(S)=\{x\in X\ |\ f(x)\in S\},\]
the set of all elements of \(X\) that get sent into \(S\) by \(f\). In our example of \(f\) in (1.1) above, we have \(f(X)=\{b,d\}\subset Y\) but \(f(X)\neq Y\); \(f^{-1}(\{b\})=\{1\}\), \(f^{-1}(\{d\})=\{2,3\}\), and \(f^{-1}(\{a\})=\emptyset.\) Despite the use of \(f^{-1}\) notation, this has nothing to do whether or not \(f\) is invertible.
A function \(f:X\to Y\) is _injective_ or _one-to-one_ if no two input values give us the same output value. More formally, \(f\) is injective if
\[f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}.\]
See Figure 1.5.
**Example 1**.: We can prove that a given function is injective using the criterion \(f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}\) by supposing that \(f(x_{1})=f(x_{2})\) and showing that \(x_{1}=x_{2}\). For example, we claim that the function \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=x^{3}-1\) is injective. To prove it, suppose \(f(x_{1})=f(x_{2})\), i.e., \(x_{1}^{3}-1=x_{2}^{3}-1\); then we have
\[x_{1}^{3}-1=x_{2}^{3}-1\iff x_{1}^{3}=x_{2}^{3}\iff x_{1}=x_{2},\]
and we have \(f(x_{1})=f(x_{2})\Rightarrow x_{1}=x_{2}\).
**Observation 1**.: Another equivalent definition of injectivity is that \(f\) is injective if and only if the preimage \(f^{-1}(\{y\})\) of every single element set has at most one element.
A function \(f:X\to Y\) is _surjective_ or _onto_ if every potential output value in \(Y\) gets hit by \(f\). More formally, \(f\) is surjective if
\[y\in Y\Rightarrow y=f(x)\text{ for some }x\in X.\]
In particular, \(f\) is surjective if \(f(X)=Y\), i.e., if the image and codomain are the same. See Figure 1.6.
Figure 1.5: Injectivity
**Observation 2**.: A function \(f:X\to Y\) is surjective if and only if no \(f^{-1}(\{y\})\) is empty, or equivalently \(f^{-1}(S)=\emptyset\) implies \(S=\emptyset\).
A function that is both injective and surjective is called _bijective_; a bijective function is called a _bijection_. A bijection is essentially a relabeling of elements of \(Y\) with labels in \(X\). Note that functions in general can be injective, surjective, bijective, or neither.
**Example 2**.: For any set \(X\), the _identity function_ on \(X\), \(\operatorname{Id}_{X}:X\to X\), is given by \(\operatorname{Id}_{X}(x)=x\) for all \(x\in X.\) That is, \(\operatorname{Id}_{X}\) is the function that just hands you back whatever input value you give it as output. The identity function is a bijection.
If \(f:X\to Y\) is a bijection, then for every \(y\in Y\) the preimage set \(f^{-1}(\{y\})\) is nonempty and contains a unique element \(x\); in this situation, we can define the _inverse function_\(f^{-1}:Y\to X\) by setting \(f^{-1}(y)=x\), where \(f^{-1}(\{y\})=\{x\}\).
If \(f:X\to Y\) and \(g:Y\to Z\), there is a function \(g\circ f:X\to Z\) called the _composite_ of \(f\) and \(g\) defined by \((g\circ f)(x)=g(f(x))\). See Figure 1.7.
**Example 3**.: For functions given by formulas, composing \(f(x)\) with \(g(x)\) means evaluating \(g\) at \(f(x)\), i.e., “plugging in” \(f(x)\) into \(g(x)\), so that
\[(g\circ f)(x)=g(f(x)).\]
For instance, if \(g(x)=x^{2}+3x-1\) and \(f(x)=x+1\), then
\[g(f(x)) =f(x)^{2}+3f(x)-1=(x+1)^{2}+3(x+1)-1\] \[=x^{2}+2x+1+3x+3-1=x^{2}+5x+3.\]
Figure 1.7: Composition of \(g\) by \(f\)
Figure 1.6: Surjectivity
**Example 4**.: Suppose \(f:X\to Y\) and \(g:Y\to X\) satisfy \((g\circ f)(x)=x\) and \((f\circ g)(y)=y\) for all \(x\in X\) and \(y\in Y\). Then we say \(g\) is the _inverse function_ of \(f\) and write \(g=f^{-1}\). Note that \(f:X\to Y\) has an inverse function if and only if \(f\) is bijective, i.e., if and only if the preimage set \(f^{-1}(\{y\})\) of each element of \(Y\) has exactly one element \(x\); then the inverse function simply sends each element \(y\) to its preimage value \(x\).
If \(f:X\to Y\), \(g:Y\to Z\), and \(h=g\circ f:X\to Z\), we say that \(h\)_factors through_\(g\) (or \(f\) or \(Y\)). This is often expressed in the form of a diagram, e.g., Figure 1.8.
In particular, a function \(f:X\to Y\) has an inverse if and only if the identity map \(I_{X}:\;X\to X\) factors through \(f\) (Figure 1.9).
### 1.4 Induction
One important “tool” of the trade which one encounters quite often is the use of induction arguments. Induction is used in conjunction with the natural numbers \(\mathbb{N}\) or sometimes with \(\{0\}\cup\mathbb{N}\). The general principle behind induction is as follows:
Let \(P(n)\) be a proposition about \(n\). Then \(P(n)\) is true for all \(n\in\mathbb{N}\), provided that:
* \(P(1)\) is true.
* For each \(k\in\mathbb{N}\), if \(P(k)\) is true, then \(P(k+1)\) is true.
We refer the verification of part a) above as the _basis for induction_ and part b) as the _inductive step_. The assumption that \(P(k)\) is true is known as the _induction hypothesis_. The following is a simple example of a proof by induction.
Figure 1.9: Factorization of the identity map
Figure 1.8: \(g\circ f\) factors through \(f\) or \(g\)
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