UNIVERSITI PENDIDKAN SULTAN IDRIS

PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT

SUBTOPIC 2CONDITIONAL STATEMENT

Topics

1. Conditional

3. De Morgan’s Law For Logic

2. Biconditional

1. Conditional

Propositional Logic – Implication It means that the operator that forms a sentence from two given

sentences and corresponds to the English if …then … Let p and q be propositions. The compound proposition “if p then

q“, denoted “p → q“, is false when p is true and q is false, and is true otherwise.

This compound proposition p → q is called the implication (or the conditional statement) of p and q.

p is called hypothesis ( or antecedent or premise ) and q is called the conclusion ( or consequence ).

Example 1

Example : If muzzamer is the agent of Herbalife (p), then he used the product (q). If p, then 2 + 2 = 4

Truth the table for the implication:

p q p → q

T T T

T F F

F T T

F F T

Remarks : The implication p → q is false only when p is true then q is false. The implication p → q is true when p is false whatever the truth value of q.

Implication : If p then q p implies q q is p p only if q q when p

Remarks and Implication

p is sufficient for q a sufficient condition for q is p q follows from p q whenever p

Definition : Let P and Q be two propositions. P ↔ Q is true whenever P and Q have the same truth

values. The proposition P ↔ Q is called biconditional or

equivalence, it is pronounced “P if and only if Q”.

2. Biconditional

Example :

Let ;

p : Jamal receives a scholarship

q : Jamal goes to college

The proposition can be written symbolically as p ↔ q. Since the hypothesis q is false, the conditional proposition is true.

Example 2

The converse of the propositions is :

“If Jamal goes to college, then he receives the

scholarship”. This is considered to be true precisely when p and q have

the same truth values). If p and q are propositions, the proposition

p if and only if q Is called a biconditional proposition and is denoted

p ↔ q

Example 2 cont…

Truth table for the biconditional:

Example 2 cont…

p q p ↔ q

T T T

T F F

F T F

F F T

Similarly to standard algebra, there are laws to manipulate logical expressions, given as logical equivalences.

Logical Equivalences

Commutative laws

• P V Q ≡ Q V P• P Λ Q ≡ Q Λ P

Associative laws

• (P V Q) V R ≡ P V (Q V R)

• (P Λ Q) Λ R ≡ P Λ (Q Λ R)

Distributive laws:

• (P V Q) Λ (P V R) ≡ P V (Q Λ R)

• (P Λ Q) V (P Λ R) ≡ P Λ (Q V R)

Verify the first of De Morgan’s Law

⌐ (p ˅ q) ≡ ⌐p ˄ ⌐q, ⌐ (p ˄ q) ≡ ⌐p ˅ ⌐q

By writing the truth table for P = ⌐ (p ˅ q) and Q = ⌐p ˄ ⌐q, we can verify that, given any truth values of p and q, either P or Q are both true or P and true are the both false:

3. De Morgan’s Law For Logic

Truth table for De Morgan’s Law :

Example 4

p q ⌐ (p ˅ q) ⌐p ˄ ⌐q

T T F F

T F F F

F T F F

F F T T