3. Topics
2.
1. Conditional
Biconditional
3. De Morgan’s
Law For Logic
4.
5. 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 ).
6.
7. 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
8.
9. Remarks and Implication
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 is sufficient for q
p implies q a sufficient condition for q is p
q is p q follows from p
p only if q q whenever p
q when p
10.
11. 2. Biconditional
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”.
12.
13. Example 2
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.
14.
15. Example 2 cont…
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
16.
17. Example 2 cont…
Truth table for the biconditional:
p q p↔q
T T T
T F F
F T F
F F T
18.
19. Logical Equivalences
Similarly to standard algebra, there are laws to manipulate
logical expressions, given as logical equivalences.
Commutative Distributive
Associative laws
laws laws:
• PV Q ≡ Q V P • (P V Q) V R ≡ • (P V Q) Λ (P V R)
• PΛ Q ≡ Q Λ P P V (Q V R) ≡ P V (Q Λ R)
• (P Λ Q) Λ R ≡ • (P Λ Q) V (P Λ R)
P Λ (Q Λ R) ≡ P Λ (Q V R)
20.
21. 3. De Morgan’s Law For Logic
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:
22.
23. Example 4
Truth table for De Morgan’s Law :
p q ⌐ (p ˅ q) ⌐p ˅ ⌐q
T T F F
T F F F
F T F F
F F T T
