2102 - Symbols we’ll use

Propositions
Concept	Java/C	Python	This class	Bitwise	Name	Other
true	`true`	`True`	$\top$ or $1$	`-1`	tautology	T
false	`false`	`False`	$\bot$ or $0$	`0`	contradiction	F
not $P$	`!p`	`not p`	$\lnot P$ or $\overline{P}$	`~p`	negation
$P$ and $Q$	`p && q`	`p and q`	$P \land Q$	`p & q`	conjunction	$P Q$ , $P \cdot Q$
$P$ or $Q$	`p \|\| q`	`p or q`	$P \lor Q$	`p \| q`	disjunction	$P + Q$
$P$ xor $Q$	`p != q`	`p != q`	$P \oplus Q$	`p ^ q`	parity	$P ⊻ Q$ ,
$P$ implies $Q$			$P \rightarrow Q$		implication	$P \supset Q$ , $P \Rightarrow Q$
$P$ iff $Q$	`p == q`	`p == q`	$P \leftrightarrow Q$		bi-implication	$P \Leftrightarrow Q$ , $P$ xnor $Q$

Proofs
Concept	Symbol	Meaning
equivalent	$\equiv$	$A \equiv B$ means $A \leftrightarrow B$ is a tautology
entails	$\vDash$	$A \vDash B$ means $A \rightarrow B$ is a tautology
provable	$\vdash$	$A \vdash B$ means $A$ proves $B$ ; it means both $A \vDash B$ and I know $B$ is true because $A$ is true $\vdash B$ (without $A$ ) means I know $B$ is true
therefore	$\therefore$	$\therefore A$ means the lines above this $\vdash A$ $\therefore A$ also connotes $A$ is the thing we wanted to show
proof done	∎ q.e.d.	marks the end of a written (prose) proof
hypothesis		something we expect is true
theorem		something we’ve proven is true
corollary		small theorem that builds off of the main theorem
lemma		small theorem that helps set up the proof of the main theorem

Arithmetic
Concept	Symbol	Meaning
floor	$\lfloor x \rfloor$	the largest integer not larger than $x$ $x$ rounded down to an integer
ceiling	$\lceil x \rceil$	the smallest integer not smaller than $x$ $x$ rounded up to an integer
exponent	$x^y$	$x$ multiplied by itself $y$ times
sum	$\displaystyle \sum_{x \in S} f(x)$	the sum of all members of $\{ f(x) \;\|\; x \in S\}$ By definition, 0 if $S = \{\}$
sum	$\displaystyle \sum_{x=a}^{b} f(x)$	$\displaystyle \sum_{x\in S} f(x)$ where $S = \{ x \;\|\; (x \in \mathbb Z) \land (a \le x \le b)\}$ the sum of $f(x)$ applied to integers between $a$ and $b$ inclusive
product	$\displaystyle \prod_{x \in S} f(x)$	the product of all members of $\{ f(x) \;\|\; x \in S\}$ By definition, 1 if $S = \{\}$
product	$\displaystyle \prod_{x=a}^{b} f(x)$	$\displaystyle \prod_{x\in S} f(x)$ where $S = \{ x \;\|\; (x \in \mathbb Z) \land (a \le x \le b)\}$ the product of $f(x)$ applied to integers between $a$ and $b$ inclusive
factorial	$x!$	$\displaystyle \prod_{i=1}^{x} i$ the product of all positive integers less than or equal to $x$ the number of permutations of a length- $x$ sequence with distinct members
choose	$\displaystyle {n \choose k}$	$\displaystyle {n! \over (n-k)! k!}$ the number of $k$ -member subsets of an $n$ -element set