CS128 – Discrete Mathematics for Computer Science

Dr. Daniel Tauritz (Dr. T)Department of Computer Science

tauritzd@mst.eduhttp://web.mst.edu/~tauritzd/

CS128 course website: http://web.mst.edu/~tauritzd/courses/cs128/fs2009

Propositional Logic

Definition

A statement (or proposition) is a sentence that is true or false but not both.

• Truth values

• Logical connectives

• Negation, conjunction, disjunction

• or vs. xor

• Order of operations

• Compound statements

• Statements vs. statement forms

• Truth Tables

Definition

Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables.

Definition

Two statements are called logically equivalent if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements.

De Morgan’s Laws

The negation of an and statement is logically equivalent to the or statement in which each component is negated.

The negation of an or statement is logically equivalent to the and statement in which each component is negated.

Definition

A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a tautology is a tautological statement.

Definition

A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. A statement whose form is a contradiction is a contradictory statement.

Definition

If p and q are statement variables, the conditional of q by p is “If p then q” or “p implies q”. It is false when p is true and q is false; otherwise it’s true. We call p the hypothesis (or antecedent) of the conditional and q the conclusion (or consequent).

Definition

The contrapositive of a conditional statement of the form “If p then q” is

“If ~q then ~p”

Symbolically: The contrapositive of p→q is ~q →~p

Definition

An argument is a sequence of statements and an argument form is a sequence of statement forms. All argument statements, except for the final one, are called premises (or assumptions or hypotheses); the final statement is called the conclusion.

Definition

To say that an argument form is valid means that regardless of the substituted statements, if the resulting premises are true, then the conclusion is also true.

To say that an argument is valid means that its form is valid.

Testing the validity of an argument form

1.Identify premises and conclusion

2.Construct truth table

3.If there is any row in which all the premises are true and the conclusion false, then the form is invalid; otherwise it’s valid

Tip: you only need to complete critical rows (rows whose premises are all true)

Definition

A syllogism is an argument form consisting of two premises and a conclusion.

Modus PonensIf p then q

p

tf q

Modus TollensIf p then q

~q

tf ~p

Additional rules of inference

Generalizationp q

tf p v q tf p v q

Specializationp ^ q p ^ q

tf p tf q

Predicate Logic

Definition

A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted.

Example

Let P(x) be the predicate “x3 > x” with domain the set R of all real numbers.

P(1): 1>1 False

P(-1): -1>-1 False

P(2): 8>2 True