Summary

Binary operations. A binary operation $*$ on a non-empty set $S$ assigns a unique element $a*b\in S$ to every ordered pair $(a,b)\in S\times S$ -- so every binary operation automatically satisfies closure.

Properties. $*$ is commutative if $a*b=b*a\ \forall\,a,b\in S$; associative if $(a*b)*c=a*(b*c)\ \forall\,a,b,c\in S$; has an identity $e\in S$ if $a*e=a=e*a\ \forall\,a\in S$; and, given an identity, $b\in S$ is the inverse of $a$ (written $a^{-1}$) if $a*b=e=b*a$. Theorem 12.1: the identity element, if it exists, is unique. Theorem 12.2: the inverse of an element, if it exists, is unique.

Boolean matrices. A Boolean matrix has every entry $0$ or $1$; join $A\vee B$ takes entrywise $\max$ and meet $A\wedge B$ takes entrywise $\min$ -- both closed, commutative, associative, each with an identity ($O$ for join, $U$ for meet), neither with inverses.

Modular arithmetic. For modulus $n>1$, $a\equiv b\pmod n$ means $b$ is the least non-negative remainder of $a$ on division by $n$ ($0\le b\le n-1$). Addition modulo $n$ ($+_n$) and multiplication modulo $n$ ($\times_n$) are the corresponding binary operations on $\mathbb Z_n=\{0,1,\ldots,n-1\}$.

Mathematical logic studies reasoning through mathematical symbols. A statement/proposition is a declarative sentence, true or false but not both. Negation $\neg p$ flips $p$'s truth value. Conjunction $p\wedge q$ is $T$ only when both are $T$. Disjunction $p\vee q$ is $F$ only when both are $F$. The conditional $p\to q$ has truth value $F$ only when $p$ is $T$ and $q$ is $F$. The biconditional $p\leftrightarrow q$ is $T$ exactly when $p,q$ share the same truth value.

Tautology, contradiction, contingency. A tautology ($\mathbb T$) is always $T$; a contradiction ($\mathbb F$) is always $F$; a contingency is neither. …