UNIVERSITI PENDIDKAN SULTAN IDRIS

PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT

SUBTOPIC 1PROPOSITIONS AND

LOGICAL OPERATIONS

Definition : A proposition is a declarative sentence that is either

TRUE or FALSE, but NOT BOT The example of declarative sentence that are not propositions:

1. Zulhelmy will not win the Thomas cup this year.

x + 1 = 3

2. Kuala Lumpur is a capital of Malaysia.

2 + 2 = 4

Propositional Logic

Notation

We use letters to denote propositional variables that represent propositions.

The conventional letters used to denote numerical variable. The conventional letters used for propositional variables are p ,

q , r , s ,….. The area of logic that deals with propositions is called

propositional calculus or propositional logic.

Quantifiers

1.

( 1 )

Negation

( 2)

Conjunction

( 3 )

Disjunction

( 4 )

Exclusive Disjunction

Propositional Logic

Negation is an other propositions The compound proposition is “it is not the case that p”. It called the negation of p and the denoted of ⌐p . The truth value of the negation of p is the opposite of the

truth value of p. The proposition ⌐p is also can say as “not p”.

1. Propositional Logic – Negation

Example : Discrete mathematics is Hasrul’s favorite subject. ( p ) Discrete mathematics is NOT Hasrul’s favorite subject.(⌐p )

Truth the table for negation :

The truth table presents the relations between the truth values of many propositions involved in a compound proposition

Example 1

p ⌐p

True False

False True

Conjunction is a compound proposition that has components joined by the word and or its symbol.

It is true only if both or all the components are true. Let p and q be propositions :

The compound proposition “p and q“ .

“p ˄ q”, is true when both p and q are true and

false otherwise

2. Propositional Logic –Conjunction

Example :

Faiz is a singer (p) and dancer (q).

Truth the table for conjunction :

Example 2

p q p ˄ q

T T T

T F F

F T F

F F F

Disjunction is a proposition that presents two or more alternative terms, with the assertion that only one is true.

The compound proposition “p or q“ or “p ˅ q“, is false when both p and q are false and true otherwise. The compound proposition “p ˅ q“ is called disjunction

of p and q.

3. Propositional Logic –Disjunction

Example :

Faiz is a singer (p) or dancer (q).

Truth the table for disjunction:

Example 3

P Q p ˅ q

T T T

T F T

F T T

F F F

Exclusive Disjunction is a Logic the connective. It gives the value true to a disjunction if one or other,

but not both, of the disjunction are true. The compound proposition “p exclusive or q“, denoted

“p q“, is true when exactly one of p and q is true and ⊕is false otherwise.

This compound proposition “p q“ is called exclusive ⊕disjunction of p and q.

4. Propositional Logic –Exclusive Disjunction

Example :

You can have a PS2 (p) or PSP (q).

Truth the table for the exclusive disjunction:

Example 4

p q p q⊕

T T F

T F T

F T T

F F F