OPTIMIZED LOGIC STUDY GUIDE (CH 2.2-3.4)

Conditional (p→q)

False only when p is true and q is false.

Truth table for p→q

Only false in the row where p is true and q is false.

Vacuously true

Occurs when the hypothesis p is false.

Key equivalence of p→q

p→q is equivalent to ~p∨q.

Negation of p→q

~(p→q) is equivalent to p∧~q.

Contrapositive

~q→~p, which is equivalent to the original conditional.

Converse

q→p, which is not equivalent to the original conditional.

Inverse

~p→~q, which is not equivalent to the original conditional.

Meaning of 'p only if q'

Represents the conditional p→q.

Biconditional (p↔q)

Defined as (p→q)∧(q→p).

Necessary condition

Saying 'You can't have p without q' means p→q.

Sufficient condition

Saying 'q automatically gives you p' means q→p.

Modus Ponens

If p→q and p are true, then q must be true.

Modus Tollens

If p→q and ~q are true, then ~p must be true.

Transitivity

If p→q and q→r are true, then p→r must be true.

Elimination

If p∨q and ~p are true, then q must be true.

Division into Cases

If p∨q, p→r, and q→r are true, then r must be true.

Contradiction Rule

If ~p leads to a contradiction, then p must be true.

Converse Error

The fallacy where p→q and q leads to the invalid conclusion of p.

Inverse Error

The fallacy where p→q and ~p leads to the invalid conclusion of ~q.

Predicate P(x)

A statement that contains variables.

Truth set

The set of values that make the predicate P(x) true.

Universal quantifier (∀x)

Indicates 'for all x'.

Existential quantifier (∃x)

Indicates 'there exists at least one x'.

Universal statement form

∀x, P(x) or ∀x, if P(x) then Q(x).

Negation of universal statements

~(∀x, P(x)) is equivalent to ∃x, ~P(x).

Negation of existential statements

~(∃x, P(x)) is equivalent to ∀x, ~P(x).

Negating Universal Conditional

~(∀x, P(x)→Q(x)) is equivalent to ∃x, P(x)∧~Q(x).

Order of quantifiers

The order matters: ∀x, ∃y, P(x,y) is not the same as ∃y, ∀x, P(x,y).

Negating multiple quantifiers

~(∀x, ∃y, P(x,y)) is equivalent to ∃x, ∀y, ~P(x,y).

Universal Instantiation

If ∀x, P(x) is true, then P(a) is true for any specific a.

Universal Modus Ponens

If ∀x, P(x)→Q(x) and P(a) are true, then Q(a) must be true.

Universal Modus Tollens

If ∀x, P(x)→Q(x) and ~Q(a), then ~P(a) must be true.

Venn diagram for 'All A are B'

Represented as A circle inside the B circle.

Venn diagram for 'No A are B'

Represented as non-overlapping circles.

'→' Symbol

Represents a conditional statement (if-then).

'↔' Symbol

Represents a biconditional statement (if and only if).

'∧' Symbol

Represents conjunction (and).

'∨' Symbol

Represents disjunction (or).

'~' Symbol

Represents negation (not).

'∀' Symbol

Represents the universal quantifier.

'∃' Symbol

Represents the existential quantifier.

'∈' Symbol

Represents 'element of'.

'∴' Symbol

Represents 'therefore'.

Operation order for logical expressions

Negation (~), followed by conjunction (∧) and disjunction (∨), then conditional (→) and biconditional (↔).