UNIVERSITI PENDIDKAN SULTAN IDRIS PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT
Topics 2.1. Conditional Biconditional 3. De Morgan’s Law For Logic
1. ConditionalPropositional 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 1Example : If muzzamer is the agent of Herbalife (p), then he used the product (q). If p, then 2 + 2 = 4Truth the table for the implication: p q p→q T T T T F F F T T F F T
Remarks and ImplicationRemarks : 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
2. BiconditionalDefinition : 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”.
Example 2Example :Let ; p : Jamal receives a scholarship q : Jamal goes to collegeThe proposition can be written symbolically as p ↔ q.Since the hypothesis q is false, the conditional propositionis true.
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 havethe same truth values).If p and q are propositions, the proposition p if and only if qIs called a biconditional proposition and is denoted p↔q
Example 2 cont…Truth table for the biconditional: p q p↔q T T T T F F F T F F F T
Logical EquivalencesSimilarly to standard algebra, there are laws to manipulatelogical 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)
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:
Example 4Truth 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