Conditional Statement/Implication CSCI 1900 Discrete Structures •"ifp then q" • Denoted p ⇒ q – p is called the antecedent or hypothesis Conditional Statements – q is called the consequent or conclusion Reading: Kolman, Section 2.2 • Example: – p: I am hungry q: I will eat – p: It is snowing q: 3+5 = 8 CSCI 1900 – Discrete Structures Conditional Statements – Page 1 CSCI 1900 – Discrete Structures Conditional Statements – Page 2 Conditional Statement/Implication Truth Table Representing Implication (continued) • In English, we would assume a cause- • If viewed as a logic operation, p ⇒ q can only be and-effect relationship, i.e., the fact that p evaluated as false if p is true and q is false is true would force q to be true. • This does not say that p causes q • If “it is snowing,” then “3+5=8” is • Truth table meaningless in this regard since p has no p q p ⇒ q effect at all on q T T T • At this point it may be easiest to view the T F F operator “⇒” as a logic operationsimilar to F T T AND or OR (conjunction or disjunction). F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 3 CSCI 1900 – Discrete Structures Conditional Statements – Page 4 Examples where p ⇒ q is viewed Converse and contrapositive as a logic operation •Ifp is false, then any q supports p ⇒ q is • The converse of p ⇒ q is the implication true. that q ⇒ p – False ⇒ True = True • The contrapositive of p ⇒ q is the –False⇒ False = True implication that ~q ⇒ ~p • If “2+2=5” then “I am the king of England” is true CSCI 1900 – Discrete Structures Conditional Statements – Page 5 CSCI 1900 – Discrete Structures Conditional Statements – Page 6 1 Converse and Contrapositive Equivalence or biconditional Example Example: What is the converse and •Ifp and q are statements, the compound contrapositive of p: "it is raining" and q: I statement p if and only if q is called an get wet? equivalence or biconditional – Implication: If it is raining, then I get wet. • Denoted p ⇔ q – Converse: If I get wet, then it is raining. – Contrapositive: If I do not get wet, then it is not raining. CSCI 1900 – Discrete Structures Conditional Statements – Page 7 CSCI 1900 – Discrete Structures Conditional Statements – Page 8 Equivalence Truth table Proof of the Contrapositive Compute the truth table of the statement • The only time that the expression can evaluate as (p ⇒ q) ⇔ (~q ⇒ ~p) true is if both statements, p and q, are true or both are false p q p ⇒ q ~q ~p ~q ⇒ ~p (p ⇒ q) ⇔ (~q ⇒ ~p) p Q p⇔q T T T F F T T T T T T F F T F F T T F F F T T F T T T F T F F F T T T T T F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 9 CSCI 1900 – Discrete Structures Conditional Statements – Page 10 Tautology and Contradiction Contingency • A statement that is true for all of its • A statement that can be either true or false propositional variables is called a depending on its propositional variables is tautology. (The previous truth table called a contingency was a tautology.) • Examples • A statement that is false for all of its –(p ⇒ q) ⇔ (~q ⇒ ~p) is a tautology propositional variables is called a – p ∧ ~p is an absurdity contradiction or an absurdity –(p ⇒ q) ∧ ~p is a contingency since some cases evaluate to true and some to false. CSCI 1900 – Discrete Structures Conditional Statements – Page 11 CSCI 1900 – Discrete Structures Conditional Statements – Page 12 2 Contingency Example Logically equivalent The statement (p ⇒ q) ∧ (p ∨ q) is a • Two propositions are logically equivalent contingency or simply equivalent if p ⇔ q is a tautology. p q p ⇒ q p ∨ q (p ⇒ q) ∧ (p ∨ q) • Denoted p ≡ q T T T T T T F F T F F T T T T F F T F F CSCI 1900 – Discrete Structures Conditional Statements – Page 13 CSCI 1900 – Discrete Structures Conditional Statements – Page 14 Example of Logical Equivalence Additional Properties Columns 5 and 8 are equivalent, and therefore, p (p ⇒ q) ≡ ((~p) ∨ q) “if and only if” q p q (p ⇒ q) ~p ((~p) ∨ q) (p ⇒ q) ⇔ ((~p) ∨ q) p q r q ∧ r p ∨ p ∨ q p ∨ r (p ∨ q) ∧ p ∨ (q ∧ r) ⇔ (q∧r) ( p ∨ r) ( p ∨ q) ∧ ( p ∨ r) T T T T T T T T T T T T F T T T T F F T T T T T T F F F F T T F T F T T T T T T F F F T T T T T F T T T T T F T T T T T T T T F T F F F T F F T F F T T T T F F T F F F T F T F F F F F F F F T CSCI 1900 – Discrete Structures Conditional Statements – Page 15 CSCI 1900 – Discrete Structures Conditional Statements – Page 16 Additional Properties (p ⇒ q) ≡ (~q ⇒ ~p) p q (p ⇒ q) ~q ~p (~q ⇒ ~p) (p ⇒ q) ⇔ (~q ⇒ ~p) T T T F F T T T F F T F F T F T T F T T T F F T T T T T CSCI 1900 – Discrete Structures Conditional Statements – Page 17 3.