2.8.1: Key Concepts

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Key Concepts

2.1 Statements and Quantifiers

Logical statements have the form of a complete sentence and make claims that can be identified as true or false.
Logical statements are represented symbolically using a lowercase letter.
The negation of a logical statement has the opposite truth value of the original statement.
Be able to
- Determine whether a sentence represents a logical statement.
- Write and translate logical statements between words and symbols.
- Negate logical statements, including logical statements containing quantifiers of all, some, and none.

2.2 Compound Statements

Logical connectives are used to form compound logical statements by using words such as and, or, and if …, then.
A conjunction is a compound logical statement formed by combining two statements with the words “and” or “but.” If the two independent clauses are represented by $p$ and $q$ , respectively, then the conjunction is written symbolically as $p \land q$ . For the conjunction to be true, both $p$ and $q$ must be true.
A disjunction joins two logical statements with the or connective. In, logic or is inclusive. For an or statement to be true at least one statement must be true, but both may also be true.
A conditional statement has the form if $p$ , then $q$ , where $p$ and $q$ are logical statements. The only time the conditional statement is false is when $p$ is true, and $q$ is false.
The biconditional statement is formed using the connective $if and only if$ for the biconditional statement to be true, the true values of $p$ and $q$ , must match. If $p$ is true then $q$ must be true, if $p$ is false, then $q$ must be false.

Translate compound statements between words and symbolic form.

Connective	Symbol	Name
and but	$\land$	conjunction
or	$\lor$	disjunction, inclusive or
not	~	negation
if $\dots$ , then implies	$\to$	conditional, implication
if and only if	$\leftrightarrow$	biconditional

The dominance of connectives explains the order in which compound logical statements containing multiple connectives should be interpreted.

The dominance of connectives should be applied in the following order

Parentheses
Negations
Disjunctions/Conjunctions, left to right
Conditionals
Biconditionals

A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last. — Figure 2.18

2.3 Constructing Truth Tables

Determine the true values of logical statements involving negations, conjunctions, and disjunctions.
- The negation of a logical statement has the opposite true value of the original statement.
- A conjunction is true when both $p$ and $q$ are true, otherwise it is false.
- A disjunction is false when both $p$ and $q$ are false, otherwise it is true.

Know how to construct a truth table involving negations, conjunctions, and disjunctions and apply the dominance of connectives to determine the truth value of a compound logical statement containing, negations, conjunctions, and disjunctions.

Negation		Conjunction (AND)			Disjunction (OR)
$p$	$~ p$	$p$	$q$	$p \land q$	$p$	$q$	$p \lor q$
T	F	T	T	T	T	T	T
F	T	T	F	F	T	F	T
		F	T	F	F	T	T
		F	F	F	F	F	F

A logical statement is valid if it is always true. Know how to construct a truth table for a compound statement and use it to determine the validity of compound statements involving negations, conjunctions, and disjunctions.

2.4 Truth Tables for the Conditional and Biconditional

The conditional statement, if

p

then

q

, is like a contract. The only time it is false is when the contract has been broken. That is, when

p

is true, and

q

is false.

Conditional
$p$	$q$	$p \to q$
T	T	T
T	F	F
F	T	T
F	F	T

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2.1 Statements and Quantifiers

2.2 Compound Statements