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1  Sentential Logic

Chapter 1 of How To Prove It introduces the following symbols of logic:

Symbol	Meaning
$\mathrm{\neg }$	not
$\wedge$	and
$\vee$	or
$\to$	if … then
$\leftrightarrow$	iff (that is, if and only if)

As we will see, Lean uses the same symbols, with the same meanings. A statement of the form $P\wedge Q$ is called a conjunction, a statement of the form $P\vee Q$ is called a disjunction, a statement of the form $P\to Q$ is an implication or a conditional statement (with antecedent $P$ and consequent $Q$), and a statement of the form $P\leftrightarrow Q$ is a biconditional statement. The statement $\mathrm{\neg }P$ is the negation of $P$.

This chapter also establishes a number of logical equivalences that will be useful to us later:

Name		Equivalence
De Morgan’s Laws	$\mathrm{\neg }(P\wedge Q)$	is equivalent to	$\mathrm{\neg }P\vee \mathrm{\neg }Q$
	$\mathrm{\neg }(P\vee Q)$	is equivalent to	$\mathrm{\neg }P\wedge \mathrm{\neg }Q$
Double Negation Law	$\mathrm{\neg }\mathrm{\neg }P$	is equivalent to	$P$
Conditional Laws	$P\to Q$	is equivalent to	$\mathrm{\neg }P\vee Q$
	$P\to Q$	is equivalent to	$\mathrm{\neg }(P\wedge \mathrm{\neg }Q)$
Contrapositive Law	$P\to Q$	is equivalent to	$\mathrm{\neg }Q\to \mathrm{\neg }P$

Finally, Chapter 1 of HTPI introduces some concepts from set theory. A set is a collection of objects; the objects in the collection are called elements of the set. The notation $x\in A$ means that $x$ is an element of $A$. Two sets $A$ and $B$ are equal if they have exactly the same elements. We say that $A$ is a subset of $B$, denoted $A\subseteq B$, if every element of $A$ is an element of $B$. If $P(x)$ is a statement about $x$, then $\{x\mid P(x)\}$ denotes the set whose elements are the objects $x$ for which $P(x)$ is true. And we have the following operations on sets:

$A\cap B=\{x\mid x\in A\wedge x\in B\}=$ the intersection of $A$ and $B$,

$A\cup B=\{x\mid x\in A\vee x\in B\}=$ the union of $A$ and $B$,

$A\mathrm{\setminus }B=\{x\mid x\in A\wedge x\notin B\}=$ the difference of $A$ and $B$,

$A∆B=(A\mathrm{\setminus }B)\cup (B\mathrm{\setminus }A)=$ the symmetric difference of $A$ and $B$.