UNIVERSITI PENDIDKAN SULTAN IDRIS

PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT

SUBTOPIC 4METHOD OF PROOF

3.1 Tautology and contradiction.

Tautology:

A proposition that is always true for all possible value of its propositional variables.

P ¬p p ˅ ¬p

T F T

F T T

Example of a TautologyThe compound proposition p ˅ ¬p is a tautology because it is always true.

Contradiction:

A proposition that is always false for all possible values of its propositional variables.

P ¬p p ˄ ¬p

T F F

F T F

1.

Contingency:

A proposition that can either be true or false depending on the truth values of its propositional variables.

A contingency table is a table of counts. A two-dimensional contingency table is formed by classifying subjects by two variables. One variable determines the row categories; the

other variable defines the column categories. The combinations of row and column categories are called cells. Examples include classifying subjects by sex (male/female) and smoking status (current/former/never) or by "type of prenatal care" and "whether the birth required a neonatal ICU" (yes/no). For the mathematician, a two-dimensional

contingency table with r rows and c columns is the set {xi j: i =1... r; j=1... c}.

3.2 Argument and rules of inference

Definition:An argument is a sequence of propositions written

The symbol ∴ is read “therefore.” The propositions, ,… are called the hypotheses (or premises) or the proposition q is called the conclusion. The argument is valid provide that if the proposition are all true, then q must also be true; otherwise, the argument is invalid (or a fallacy).

:

,

Example:Determine whether the argument

p → q

p ∴ q

Is valid[First solution] We construct a

truth table for all the propositions involved.

P q p → q p q

T T T T T

T F F T F

F T T F T

F F T F F

Rule of inference Name

p → q p∴ q

Modus ponens

p → q ⌐q∴ ⌐p

Modus tollens

p∴ p ˅ q

Addition

p ˄ q ∴ p

Simplification

p q∴ p ˄ q

Conjunction

p → q q → r∴ p → r

Hypothetical syllogism

p ˅ q ⌐p∴ q

Disjunctive syllogism

Represent the argument.The bug is either in module 17 or in module 81The bug is a numerical errorModule 81 has no numerical error___________________________________________∴ the bug is in module 17.

 Given the beginning of this section symbolically and show

that it is valid.If we let

p : the bug is in module 17.q : the bug is in module 81.r : the bug is numerical error.

Example 2

The argument maybe writtenp V qrr → ⌐q∴ p

From r → ⌐q and r, we may use modus ponens to conclude ⌐q. From r V q and ⌐q, we may use the disjunctive syllogism to conclude p. Thus the conclusion p follows from the hypotheses and the argument is valid.

This method is based on Modus Ponens,[(p ⇒ q) ˄ p ] ⇒ qVirtually all mathematical theorems are

composed of implication of the type, ( The are called the hypothesis or premise,

and q is called conclusion. To prove a theorem means to show the implication is a tautology. If all the are true, the q must be also true.

DIRECT PROOF

Solution:Let p: x is odd, and q: x2 is odd. We want to prove p

→ q.Start: p: x is odd→ x = 2n + 1 for some integer n→ x2 = (2n + 1)2→ x2 = 4n2 + 4n + 1→ x2 = 2(2n2 + 2n) + 1→ x2 = 2m + 1, where m = (2n2 + 2n) is an integer→ x2 is odd→ q

Example: An even number is of the form 2n where n is an integer, whereas an odd number is 2n + 1. Prove that if x is an odd integer then x2 is also odd.

Contradiction: to prove a conditional proposition p ⇒ q by contradiction, we first assume that the hypothesis p is true and the conclusion is false (p˄ ~ q). We then use the steps from the proof of ~q ⇒ ~p to show that ~p is true. This leads to a contradiction (p˄ ~ p), which complete the proof.

INDIRECT PROOF

Definition:

An indirect proof uses rules of inference on the negation of the conclusion and on some of the premises to derive the negation of a premise. This result is called a contradiction.

Proof: Assume that x is even (negation of conclusion).

Say x = 2n (definition of even).Then = (substitution)= 2n · 2n (definition of exponentiation)= 2 · 2n2 (commutatively of multiplication.)Which is an even number (definition of even)This contradicts the premise that is odd.

Example: A theoremIf is odd, then so is x.