Section A.1 Notation and Assumed Background Knowledge
It will be advantageous for us all to agree on some language and notation.
We assume familiarity with the set \(\N\) of natural numbers, the set \(\Z\) of integers, the set \(\Q\) of rational numbers, and the set \(\R\) of real numbers. Unless otherwise specified, the symbols \(n,\,m,\,p\) will always represent elements of \(\N\) and the symbols \(x,y,z\) will always represent real numbers. Any universal set \(U\) is assumed nonempty unless otherwise indicated. We will also use the following notation:
| \(:=\) | ’’is defined as’’ |
| \(\forall\) | ’’for all’’ |
| \(\exists\) | ’’there exists’’ |
| \(\ni,\,|\) | ’’such that’’ |
| WTS | ’’want to show that’’ |
| LHS, RHS | ’’left-hand side’’, ’’right-hand side’’ |
| \(\blacksquare\) | ’’the argument/calculation/solution is complete’’ |
A quantifier such as \(\forall\) and \(\exists\) is followed by a variable \(x\) and then a proposition \(P(x)\text{.}\) Quantified statements like \(\forall x,\, P(x)\) and \(\exists x,\, P(x)\) classify the number of values of \(x\) in the domain for which \(P(x)\) is true.
The statement \(\fa x,\, P(x)\) indicates that \(P(x)\) is true for every \(x\) in the domain, as in \(\forall x\in[0,2\pi],\, \sin x\in[-1,1]\text{.}\)
The statement \(\ex x\ni P(x)\) indicates that \(P(x)\) is true for at least one (and possibly all) of the values of \(x\) in the domain, as in \(\ex x\in[0,2\pi],\ni \sin x=0\text{.}\)
When \(P(x)\) is not true for any of the values of \(x\) in the domain, we may write \(\fa x,\,\neg P(x)\) which indicated that \(P(x)\) is false for all values of \(x\) in the domain, as in \(\fa x\in\R,\,\neg(x^2\lt0)\text{,}\) or more simply \(\fa x\in\R,\,x^2\ge0\text{.}\)
Definition A.1.3.
(Set-Builder Expression) Let \(S\) be a set. If there is a propositional function \(P\) on \(U\) for which \(S\) is the truth set of \(P\text{,}\) then we write \(S\) in the set-builder expression
\begin{equation*}
S=\left\{x\in U\left|\right.P(x)\right\}\text{.}
\end{equation*}
In this notation the expression \(P(x)\) is called the elementhood test. We will not use the expression \(S=\{x\in U\,|x\in S\}\) since it provides no insight and is obnoxious.
Example A.1.4.
The set-builder expression for the set of zeros of \(\cos x\) is
\begin{equation*}
\left\{x\in\R\,\left|\,x=\frac{(2n+1)\pi}{2}\right.\text{ for some }n\in\Z\right\}
\end{equation*}
or
\begin{equation*}
\left\{x\in\R\,\left|\,x\text{ is an odd multiple of }\frac{\pi}{2}\right.\right\}\text{.}
\end{equation*}
The expression \(\left\{x\in\R\,\left|\,\cos x=0\right.\right\}\) is also “legal” but not particularly descriptive, and useful only as a starting point in a chain of further set-builder expression equalities.
Definition A.1.5.
(Element Form Expression) An alternate method of describing some sets \(S\text{,}\) is the element form expression of a set. The set \(S\) has an element form expression when \(S\) can be parametrized by letting a variable \(x\) vary through some domain \(U\text{,}\) computing \(f(x)\) for each such \(x\text{,}\) and collecting the \(f(x)\)’s into the set \(S\text{.}\) The element form expression of such a set \(S\) is
\begin{equation*}
S=\{f(x)\,|\,x\in U\}.
\end{equation*}
Example A.1.6.
The element form expression for the set of zeros of \(\cos x\) is
\begin{equation*}
\left\{\left.\frac{(2n+1)\pi}{2}\,\right|\,n\in\Z\right\}\text{.}
\end{equation*}
Remark A.1.7.
All sets have a set-builder expression but not all sets have an element form expression. For example, consider the open interval \((0,1)=\left\{x\in\R\,\left|\,0\lt x\lt1\right.\right\}\) but for which no element form expression exists.
Remark A.1.8.
The following list of assumed background knowledge(or ABK’s) will serve as a set of mathematical facts available to us. In computations, these facts will only lie in the background, but when we are constructing proofs we will cite them by number as necessary. For example, if we need the fact that the set of rational numbers \(\Q\) is closed under addition, we cite ABK 2b.
ABKs:
- Arithmetic. (Examples: \(1 + 1 = 2\text{;}\) “minus times minus is plus” and other rules of signs; dividing is the same as multiplying by the reciprocal; subtracting is adding the opposite; common denominators; and so on)
-
Closure:
-
The set \(\R\) of real numbers is closed under addition, subtraction and multiplication: \(a,b\in\R\Ra a\pm b\in\R\) and \(ab\in\R\text{.}\)The set \(\R^*:=\R\setminus\{0\}\) of nonzero real numbers is closed under division: \(a,b\in\R^*\Ra a\div b\in\R^*\text{.}\)
-
The set \(\Q\subset\R\) of rational numbers is closed under addition, subtraction and multiplication: \(a,b\in\Q\Ra a\pm b\in\Q\) and \(ab\in\Q\text{.}\)The set \(\Q^*:=\Q\setminus\{0\}\) of nonzero rational numbers is closed under division: \(a,b\in\Q^*\Ra a\div b\in\Q^*\text{.}\)
-
The set \(\Z\subset\Q\) of integers is closed under addition, subtraction and multiplication: \(a,b\in\Z\Ra a\pm b\in\Z\) and \(ab\in\Z\text{.}\)The set \(\Z^*:=\Z\setminus\{0\}\) of nonzero integers is closed under division: \(a,b\in\Z^*\Ra a\div b\in\Z^*\text{.}\)
- The set \(\N\subset\Z\) of natural numbers is closed under addition and multiplication only: \(a,b\in\N\Ra a+b\in\N\) and \(ab\in\N\text{.}\)
In general, subsets of \(\R,\Q,\Z\) and \(\N\) are not necessarily closed under the various arithmetical operations. For example \([1,3]\subset\R\text{,}\) and \(2,3\in[1,3]\) but \(2\cdot3=6\not\in[1,3]\text{.}\) -
- \(+\) and × are commutative on \(\R\text{.}\)
- \(+\) and × are associative on \(\R\text{.}\)
- The distributive property holds on \(\R\text{.}\)
- Adding, subtracting, and multiplying by, the same real number on both sides of an equality preserves the equality.
- Dividing both sides of an equality by the same nonzero real number preserves direction of the equality.
- Adding or subtracting the same real number on both sides of an inequality preserves the direction of the inequality.
- Multiplying or dividing both sides of an inequality by the same positive real number preserves the direction of the inequality. (Example: If \(x \lt 2\text{,}\) then \(3x \lt 6\text{.}\))
- Multiplying or dividing both sides of an inequality by the same negative real number reverses the direction of the inequality. (Example: If \(x \lt 2\text{,}\) then \(-3x > -6\text{.}\))
- Transitivity of equality and inequality: If \(a \ge b\) and \(b \ge c\text{,}\) then \(a \ge c\text{.}\) Same for \(\le\text{,}\) \(=\text{,}\) \(\gt\text{,}\) and \(\lt\text{.}\)
- Law of Trichotomy: If \(a\) and \(b\) are real numbers, then exactly one of the following is true: \(a=b\text{,}\) \(a\lt b\text{,}\) or \(a>b\text{.}\)
- Substitution: If \(a,b\in\R\) with \(a=b\text{,}\) then we may substitute \(b\) for \(a\) in any given expression without disrupting any relation (for example \(=,\lt,\ge\)) containing the given expression.
- If \(a\) is a real number and \(n\) is an odd natural number, then \(a\) has a unique \(n^{th}\) root in the real numbers.
- For any real numbers \(a\) and \(b\text{,}\) \(|ab|=|a||b|\text{.}\)
- For any real number \(c\text{,}\) for any positive real number \(d\text{,}\) \(|c| \lt d\) if and only if \(-d \lt c \lt d\text{.}\) This also holds when \(\lt\) is replaced by \(\leq\text{.}\)
- The quadratic formula: \(ax^2+bx+c=0\Leftrightarrow x=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}.\)
- Domains and ranges of the elementary functions from algebra and trigonometry (rational, exponential, logarithm, sine, cosine, etc.). Trig, logarithm, and exponential identities.
- The real number 0 is the identity for addition over any subset of \(\R, \Q, \C\text{.}\) The real number 1 is the identity for multiplication over any subset of \(\R, \Q, \C\text{.}\)
