CSE 311 Reference Sheets: Key Definitions and Results

$\newcommand{\T}{\mathsf{T}} \newcommand{\F}{\mathsf{F}}$

$\newcommand{rule}[2]{\begin{array}{c}#1\\ \hline \therefore #2\end{array}}$ $\newcommand{ruletemplate}[2]{\begin{array}{c}#1\\ \hline #2\end{array}}$ $\newcommand{hilite}[2]{\color{#1}{#2}}$ $\newcommand{arbitrary}[1]{\color{SteelBlue}{#1}}$ $\newcommand{specific}[1]{\color{MediumSeaGreen}{#1}}$ $\newcommand{qed}{\square}$

Topics

Logical equivalences: List of basic equivalences in propositional logic.
Boolean algebra identities: List of basic identities in boolean algebra.
Inference rules: List of inference rules for propositional and predicate logic.
Set theory definitions: List of definitions related to set theory.

Logical equivalences

List of basic equivalences in propositional logic.

Basic equivalences in propositional logic

DeMorgan’s laws: $\neg(p \wedge q) \equiv \neg p \vee \neg q$; $\neg(p \vee q) \equiv \neg p \wedge \neg q$
Law of implication: $p \rightarrow q \equiv \neg p \vee q$
Contrapositive: $p \rightarrow q \equiv \neg q \rightarrow \neg p $
Biconditional: $p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p)$
Double negation: $p \equiv \neg \neg p$

Identity: $p \wedge \mathsf{T} \equiv p$; $p \vee \mathsf{F} \equiv p$
Domination: $p \wedge \mathsf{F} \equiv \mathsf{F}$; $p \vee \mathsf{T} \equiv \mathsf{T}$
Idempotence: $p \wedge p \equiv p$; $p \vee p \equiv p$
Commutativity: $p \wedge q \equiv q \wedge p$; $p \vee q \equiv q \vee p$

Associativity: $(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$; $(p \vee q) \vee r \equiv p \vee (q \vee r)$
Distributivity: $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$; $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$
Absorption: $p \wedge (p \vee q) \equiv p$; $p \vee (p \wedge q) \equiv p$
Negation: $p \wedge \neg p \equiv \mathsf{F}$; $p \vee \neg p \equiv \mathsf{T}$

Boolean algebra identities

List of basic identities in boolean algebra.

Boolean algebra axioms and (some) theorems

Closure: $a + b \in B$; $a \cdot b \in B$
Commutativity: $a + b = b + a$; $a \cdot b = b \cdot a$
Associativity: $a + (b + c) = (a + b) + c$; $ a \cdot (b \cdot c) = (a \cdot b) \cdot c$

Distributivity: $a + (b \cdot c) = (a + b) \cdot (a + c)$; $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
Identity: $a + 0 = a$; $a \cdot 1 = a$
Complementarity: $a + a’ = 1$; $a \cdot a’ = 0$

Null: $a + 1 = 1$; $a \cdot 0 = 0$
Idempotency: $a + a = a$; $a \cdot a = a$
Involution: $(a’)’ = a$

Uniting: $a \cdot b + a \cdot b’ = a$; $(a + b) \cdot (a + b’) = a$
Absorption: $a + a \cdot b = a$; $a \cdot (a + b) = a$; $(a + b’) \cdot b = a \cdot b$; $(a \cdot b’) + b = a + b$

Factoring: $(a + b)\cdot (a’ + c) = a \cdot c + a’ \cdot b$; $a \cdot b + a’ \cdot c = (a + c) \cdot (a’ + b)$
Consensus: $(a \cdot b) + (b \cdot c) + (a’ \cdot c) = a \cdot b + a’ \cdot c$; $(a + b) \cdot (b + c) \cdot (a’ + c) = (a + b) \cdot (a’ + c)$
DeMorgan’s: $(a + b + \ldots)’ = a’ \cdot b’ \cdot \dots$; $(a \cdot b \cdot \ldots)’ = a’ + b’ + \dots$

Inference rules

List of inference rules for propositional and predicate logic.

Inference rules for propositional and predicate logic

Intro $\wedge$ $\rule{A; B}{A \wedge B}$
Elim $\wedge$ $\rule{A\wedge B}{A,B}$

Intro $\vee$ $\rule{A}{A\vee B, B\vee A}$
Elim $\vee$ $\rule{A\vee B; \neg A}{B}$

Direct Proof Rule $\rule{A\implies B}{A\rightarrow B}$
Modus Ponens $\rule{A ; A\rightarrow B}{B}$
Excluded Middle$\rule{}{A\vee\neg A}$

Elim $\forall$ $\rule{\forall x. P(x)}{P(a) \text{ for any } a}$

Intro $\forall$ $\rule{P(a); a \text{ is } \color{MediumVioletRed}{\text{arbitrary}}}{\forall x. P(x)}$

The name $a$ stands for an arbitrary value in the domain. No other name in $P$ depends on $a$.

Intro $\exists$ $\rule{P(c) \text{ for some } c}{\exists x. P(x)}$

Elim $\exists$ $\rule{\exists x. P(x)}{P(c) \text{ for a } \color{MediumVioletRed}{\text{specific }} c }$

The name $c$ is fresh and stands for a value in the domain where $P(c)$ is true. List all dependencies for $c$.

Set theory definitions

List of definitions related to set theory.

Set definitions and operations

Common sets: $\mathbb{N}=\{0,1,2,\ldots\}$ is the set of natural numbers.; $\mathbb{Z}=\{\ldots, -2, -1, 0,1,2,\ldots\}$ is the set of integers.; $\mathbb{Q}=\{\frac{p}{q} : p,q\in\mathbb{Z} \wedge q \neq 0 \}$ is the set of rational numbers.; $\mathbb{R}$ is the set of real numbers.
Containment, equality, and subsets: $x \in A$ means that $x$ is an element of $A$.; $x \not\in A$ means that $x$ is not an element of $A$.; $A\subseteq B$ means that $A$ is a subset of $B$: $\forall x. x\in A \rightarrow x\in B$.; $A=B$ means that $A$ and $B$ have the same elements: $\forall x. x\in A \leftrightarrow x\in B$.
Set operations: $A\cup B = \{x : (x\in A)\vee(x\in B)\}$ is the union of $A$ and $B$.; $A\cap B = \{x : (x\in A)\wedge(x\in B)\}$ is the intersection of $A$ and $B$.; $A\setminus B = \{x : (x\in A)\wedge(x \not\in B)\}$ is the difference of $A$ and $B$.; $A\oplus B = \{x : (x\in A)\oplus(x\in B)\}$ is the symmetric difference of $A$ and $B$.; $\overline{A} = \{x : x\not\in A\} = \{x:\neg(x\in A)\}$ is the complement of $A$ with respect to a universe $U$.
Set constructions: $S = \{ x : P(x)\}$ means that $S$ is the set of all $x$ in the domain of $P$ for which $P(x)$ is true.; $A\times B = \{(a,b): a\in A \wedge b \in B\}$ is the cartesian product of $A$ and $B$.; $\mathcal{P}(A) = \{B : B\subseteq A\}$ is the power set of $A$.; $[n] = \{1,2,\ldots,n\}$ is the set of natural numbers from 1 to $n$.