They are logically equivalent. p ? q ? ¬q ? ¬p. p q p ? q. T T. T. T F. F.
On voit que la proposition logique composée P ? Q est toujours vraie pour tous les valeurs de Propositions logiquement équivalents à l'implication.
The converse of a conditional proposition p ? q is the proposition q ? p. As we have seen the bi- conditional proposition is equivalent to the conjunction of
Show that (p ? q) ? (q ? p) is logically equivalent to p ? q. Solution 1. Show the truth values of both propositions are identical. Truth Table: p q p ?
Instead it applies to a single (possibly compound) statement. Negation has precedence over logical connectives. Thus ¬p ? q means. (¬p) ? q. The negation of
(~P ? Q) ? P; Biconditional -parentheses added by dominance of connectives (Hint: Use the fact that p ? q is equivalent to ~p ? q.) Problem: ~P ? Q.
C'est la base du raisonnement “par contraposée". 1.5 Equivalence. ? la négation de "P et Q sont équivalentes" est "l'une des propositions est vraie
P ? (Q ? R) is equivalent to (P ? Q) ? R. Idempotent Laws. P ? P is equivalent to P. P ? P is equivalent to P. Distributive Laws.
Suppose P Q
Equivalence. Description. Modus Ponens. (p ? q) p. ? q if p then q; p; ¬q. ? ¬p if p then q; not q; therefore not p. Hypothetical Syllogism. (p ? q).
p q p ? q ’ ’ & ’ & & Two formulas that are syntactically identical are also equivalent These two formulas are syntactically di?erent but have the same
Since the truth values for:(p!q) andp^:qare exactly the same for all possiblecombinations of truth values of pandq the two propositions are equivalent Solution 2 We consider how the two propositions couldfail to be equivalent Thiscan happen only if the rst is true and the second is false or vice versa Case 1
The proposition p ? q read “p if and only if q” is called bicon-ditional It is true precisely when p and q have the same truth value i e they are both true or both false 1 1 4 Logical Equivalence Note that the compound proposi-tions p ? q and ¬p?q have the same truth values: p q ¬p ¬p?q p ? q T T F T T T F F F F F T T
Two compound propositions p and q are logically equivalent if p ? q is a tautology ! Notation: p ? q ! De Morgan’s Laws: • ¬
Example Show that P ? Qand ¬P? Qare logically equivalent P Q P ? Q ¬P ¬P? Q T T T F T T F F F F F T T T T F F T T T Since the columns for P ? Q and ¬P ? Q are identical the two statements are logically equivalent This tautology is called Conditional Disjunction You can use this equivalence to replace a conditional by a
p and q have the same truth value p q Two statements are equivalent if they have the same truth value in all cases Variations of the Conditional Statement p ? q • p ? q is equivalent to q ? p the contrapositive: p ? q q ? p • p ? q is NOT equivalent to q ? p the converse