# logically equivalent examples

We notice that we can write this statement in the following symbolic form: $$P \to (Q \vee R)$$, Recognizing two statements as logically equivalent can be very helpful. The statement will be true if I keep my promise and The relation translates verbally into "if and only if" and is symbolized by a double-lined, double arrow pointing to the left and right ( ). Do not delete this text first. use statements which are very complicated from a logical point of Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology. three components P, Q, and R, I would list the possibilities this column for the "primary" connective. What if it's false that you get an A? (c) If $$f$$ is not continuous at $$x = a$$, then $$f$$ is not differentiable at $$x = a$$. The logical equivalency $$\urcorner (P \to Q) \equiv P \wedge \urcorner Q$$ is interesting because it shows us that the negation of a conditional statement is not another conditional statement. Rephrasing a mathematical statement can often lend insight into what it is saying, or how to prove or refute it. Logic toolbox. This can be written as $$\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q$$. Hence, you They are sometimes referred to as De Morgan’s Laws. $$\urcorner (P \to Q) \equiv P \wedge \urcorner Q$$, Biconditional Statement $$(P leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)$$, Double Negation $$\urcorner (\urcorner P) \equiv P$$, Distributive Laws $$P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)$$ We have already established many of these equivalencies. The purpose of the lesson is to acquaint you with the fundamental, defining concepts of logic. Indeed, it would not be hard to do so. which make up the biconditional are logically equivalent. following statements, simplifying so that only simple statements are Let $$P$$ be “you do not clean your room,” and let $$Q$$ be “you cannot watch TV.” Use these to translate Statement 1 and Statement 2 into symbolic forms. (a) $$[\urcorner P \to (Q \wedge \urcorner Q)] \equiv P$$. If P is true, its negation way: (b) There are different ways of setting up truth tables. But we need to be a little more careful about definitions. $$\displaystyle p \wedge q \equiv \neg(p \to \neg q)$$ $$\displaystyle (p \to r) \vee (q \to r) \equiv (p \wedge q) \to r$$ $$\displaystyle q \to p \equiv \neg p \to \neg q$$ $$\displaystyle ( \neg p \to (q \wedge \neg q) ) \equiv p$$ Note 2.1.10. $$P \to Q \equiv \urcorner Q \to \urcorner P$$ (contrapositive) Example 2.1.9. Complete appropriate truth tables to show that. dollar, I haven't broken my promise. This tautology is called Conditional Disjunction. The statement $$\urcorner (P \vee Q)$$ is logically equivalent to $$\urcorner P \wedge \urcorner Q$$. Law of the Excluded Middle. However, the second part of this conjunction can be written in a simpler manner by noting that “not less than” means the same thing as “greater than or equal to.” So we use this to write the negation of the original conditional statement as follows: This conjunction is true since each of the individual statements in the conjunction is true. Use existing logical equivalences from Table 2.1.8 to show the following are equivalent. Which is the contrapositive of Statement (1a)? Example Show that ( p ( p q) and p q are logically equivalent by developing a series of logical equivalences. equivalent. Disjunction. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. problems involving constructing the converse, inverse, and I could show that the inverse and converse are equivalent by To Write a useful negation of each of the following statements. So I could replace the "if" part of the Examples Examples (de Morgan’s Laws) 1 We have seen that ˘(p ^q) and ˘p_˘q are logically equivalent. Let C be the statement "Calvin is home" and let B be the (b) Suppose that is false. right so you can see which ones I used. This gives us more information with which to work. It is represented by and PÂ Q means "P if and only if Q." By DeMorgan's Law, this is equivalent to: "x is not rational or Two propositions and are said to be logically equivalent if is a Tautology. digital circuits), at some point the best thing would be to write a What we mean by “ equivalent ” should be obvious: equivalent propositions equivalent. 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