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
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