Ch. 2 Key Concepts - Contemporary Mathematics

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
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

The biconditional statement,

p

if and only if

q

, it true whenever

p

and

q

have matching true values, otherwise it is false.

Biconditional
$p$	$q$	$p \leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

Know how to construct truth tables involving conditional and biconditional statements.
Use truth tables to analyze conditional and biconditional statements and determine their validity.

2.5 Equivalent Statements

Two statements $p$ and $q$ are logically equivalent if the biconditional statement, $p \leftrightarrow q$ is a valid argument. That is, the last column of the truth table consists of only true values. In other words, $p \leftrightarrow q$ is a tautology. Symbolically, $p$ is logically equivalent to $q$ is written as: $p \equiv q .$
A logical statement is a tautology if it is always true.
To be valid a local argument must be a tautology. It must always be true.
Know the variations of the conditional statement, be able to determine their truth values and compose statements with them.

The converse of a conditional statement, if $p$ then $q$ , is the statement formed by interchanging the hypothesis and conclusion. It is the statement if $q$ then $p$ .
The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement.

The contrapositive negates and interchanges the hypothesis and the conclusion.

				Conditional	Contrapositive	Converse	Inverse
$p$	$q$	$~ p$	$~ q$	$p \to q$	$~ q \to ~ p$	$q \to p$	$~ p \to ~ q$
T	T	F	F	T	T	T	T
T	F	F	T	F	F	T	T
F	T	T	F	T	T	F	F
F	F	T	T	T	T	T	T

The conditional statement is logically equivalent to the contrapositive.
The converse is logically equivalent to the inverse.
Know how to construct and use truth tables to determine whether statements are logically equivalent.

2.6 De Morgan’s Laws

De Morgan’s Law for the negation of a disjunction states that, $~ (p \lor q)$ is logically equivalent to $~ p \land ~ q .$
De Morgan’s Law The negation of a conjunction states that, $~ (p \land q) \equiv ~ p \lor ~ q .$
Use De Morgan’s Laws to negate conjunctions and disjunctions.
The negation of a conditional statement, if $p$ then $q$ is logically equivalent to the statement $p$ and not $q$ . Use this property to write the negation of conditional statements.
Use truth tables to evaluate De Morgan’s Laws.

2.7 Logical Arguments

A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true.
The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, $p \to q$ is true and the hypothesis, $p$ is true, then the conclusion $q$ must also be true.

Law of Detachment

Premise: $p \to q$

Premise: $p$

Conclusion: $∴ q$
Know how to apply the law of detachment to determine the conclusion of a pair of statements.
The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, $p \to q$ is true and the negation of the conclusion, $~ q$ is true, then the negation of the hypothesis $~ p$ must also be true.

Law of Denying the Consequent

Premise: $p \to q$

Premise: $~ q$

Conclusion: $∴ ~ p$
Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements.
The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, $p \to q$ and $q \to r$ , then it follows that $p \to r .$

Chain Rule for Conditional Arguments

Premise: $p \to q$

Premise: $q \to r$

Conclusion: $∴ p \to r$
Know how to apply the chain rule to determine valid conclusions for pairs of true statements.

Law of Detachment
Premise:	$p \to q$
Premise:	$p$
Conclusion:	$∴ q$

Law of Denying the Consequent
Premise:	$p \to q$
Premise:	$~ q$
Conclusion:	$∴ ~ p$

Chain Rule for Conditional Arguments
Premise:	$p \to q$
Premise:	$q \to r$
Conclusion:	$∴ p \to r$