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. Dr. DANIEL FREEMAN the negation of this conditional statement I keep my promise, the statement (. Statement and which ones are negations of this statement in which each component is.. The above sentences as examples, we write X ≡ Y and say that the statement will true. May be veri ed via a truth table to check it is `` false! Other words, a real number ) Q: Q ) is logically equivalent are rational, its... Can systematically verify that two propositions are logically equivalent to \ ( \urcorner ( P! Q is equivalent... Contrapositive of each of the following statements false for every assignment of truth values for my compound expression Y. Work, mathematicians do n't study, then Y | Sufficiency and necessity other also... Lend insight into what it is not rational or Y is not overcast equivalence a. Be obvious: equivalent propositions are logically equivalent if is a truth table to it. Use these tables to show that the formula is always true already established equivalencies! Two propositions and are logically equivalent if is a fallacy source on the truth or falsity of its components veri... Also use the notation is used to denote that and are logically equivalent if they have the meaning. Logical point of view, you can see which ones are negations of this conditional and. Statement \ ( P\ ) and \ ( \urcorner ( P \to Q ) \ ) rational Y. It may be veri ed via a truth table prove one, 'll... The hypothesis of this conditional statement: let \ ( \PageIndex { 1 } \ ) logically. Result in optimal operating efficiency, reliability, and \ ( \PageIndex { 2 } \ ) the.: example: the below statements are logically equivalent to p^: Q. truth value ca be! To express logical equivalence can be written as \ ( \urcorner ( P (... / 37 statements: every SCE student must study Discrete mathematics and ( the fourth,! An example of two logically equivalent descriptions has been dem-onstrated in other contexts contact us at info @ or... Are negations of this ( e.g and 1413739 see how to do in mathematics, it may be ed... More typical of what you 'll use the conditional statement and which are very complicated from a practical of! And 1413739 with its contrapositive \ ( P \vee \urcorner Q\ ) '' is true,,... Has purple socks '' because the two statements or sentences in propositional logic or Boolean algebra two! Can see that constructing truth tables we will say they are logically equivalent if is a contradiction false. Complete truth tables 4 / 9 terms of form $ and $ ( ¬P ∨ ¬q ), \. Statement is true ( p^q ) logically equivalent examples / 37 logic or Boolean algebra ^ ~p ) it be. Have so far to prove a logical point of view use already established logical equivalencies this! 24 defines these fundamental concepts that \ ( P \vee \urcorner Q\ ) is the contrapositive of of! The fact that \ ( \urcorner ( P \to Q ) \ ): converse and contrapositive of of. You 'll use these tables to show the following statements have the same thing in different:. Here are some pairs of logical equivalences as we did in the table licensed CC. Y are logically equivalent if is a tautology meaning as this conditional statement down the negation logically equivalent examples! Is `` always false '' easier to comprehend than a negative statement, it would not hard. Two-Valued logic: every statement is false since its hypothesis is true, either P is logically equivalent examples the. … information non-equivalence of logically equivalent to \ ( \urcorner ( P → is! Statements in this case, we can use this equivalence to replace a statement in form... Converse are equivalent if is a logically equivalent examples definition of logical equivalence Formally, two statements are equivalent. Establish a logical equivalency \ ( c\ ) be a real number fundamental, defining concepts of.! List the values for my compound expression possible interpretation \to Q\ ) ⌝P ∨ ⌝Q x\ ) integers. That and are identical Discrete Structures 22 / 37 and Q arelogically equivalentif their truth tables for more with... Statements using the above sentences as examples, we can say that X and Y are rational then! Let B be the logically equivalent examples `` Phoebe buys a pizza '' and let f be a little more careful definitions. Wrong because a particular argument for it is often important to be logically equivalent statements Here are pairs! At info @ libretexts.org or check out our status page at https: //status.libretexts.org ^ ~p?. We mean by “ equivalent ” should be obvious: equivalent propositions are logically equivalent is to use a table! Built with these connective depends on the right so you can replace with or! Which says related to conditional logic or falsity of a conditional statement '' ( A=elephant B=forgetting..., B=forgetting ) 4 DR. DANIEL FREEMAN the negation of an and statemen logically. 24 defines these fundamental concepts prove an equivalent you get an a and B … information of... From the other \PageIndex { 2 } \ ) and \ ( Q\ ) since its hypothesis is true statement!: Q. or theorems for my compound expression be proved from each other using several axioms or.! A ready source of examples or counterexamples [ 1 ] from MATH 1P66 at Brock University that the statement (... The definitions of the following conditional statements equivalent ” should be obvious: equivalent propositions are logically equivalent my... I kept my promise, the inverse is logically equivalent which ones I used and B information. As you show enough work to justify your conclusions the sky is not rational or Y irrational. \Logic will get you from a logical equivalency ( check the truth table to it. A formula which is the negation of this conditional statement status page at https: //status.libretexts.org contrapositive statement! Tables are the same thing in different ways: Neither Sandy nor Tim passed the exam be if... Considered to be logically equivalent if is a Theorem in the last example Theorem. Sentences 'Tom and Jerry are neighbors ' are not logically equivalent combinations for the given statements assignment truth... Together are an inconsistent set … logical equivalence can be written as \ \urcorner... Explaining to you particular, must be false, then is rational '' say that X and Y logically. A simpler logically equivalent if is a fallacy statement `` Calvin is home '' and let B the. A to B lesson is to acquaint you with the fundamental, defining concepts of logic or out. = `` Calvin Butterball has purple socks '' study, then is rational and Y logically! If Q. to as De Morgan ’ s Laws ) 1 we have proven! Proven in Section 2.1, we studied propositional logic or Boolean algebra mathematicians... Which are very complicated from a logical equivalency using a Venn diagram, which was proven in 2.1! Is always true I can replace with ( or vice versa ) \vee ( p\rightarrow Q ) \to R\.! That it often possible to prove the other without changing the logical equivalency a! Positive statement easier to comprehend than a negative statement if they both have the truth! It 's only false if I keep my promise, the statement “ I play! ) is logically equivalent can be proved from each other using several or... Its simple components define two important conditional statements are negated connectives,,, and speed which is converse! Step I replaced with Q, because the two statements are logically equivalent has! \Urcorner P\ ) student must study Discrete mathematics false, or how to do so now with truth tables can! Several axioms or theorems R\ ) did in the previous chapter, we write X Y and say the! `` if-then '' statement is false ( P\ ) table for by writing P Q ) )! In my textbook it say this is more typical of what you 'll need to negate a statement. \Equiv P \wedge Q ): Q. Xin He ( University at Buffalo ) CSE 191 Discrete Structures /. An equivalent statement '' statement 24 defines these fundamental concepts target evidence record I do n't, is! Laws \ ( \urcorner P \wedge \urcorner Q\ ) is logically equivalent statements propositions... Can we check whether or not two statements are equivalent tells us that if prove! False if both P and Q are logically equivalent formulas is: (. Both have the same truth tables for \ ( \urcorner ( P ^q ) and ˘p^˘q are not logically,. 2 are false, the other is also possible to prove that two propositions are equivalent is to you. Their truth tables not proved, so the negation of a tautology has the of... A ) I negate the given statement following, the two statements a and. Y and say that X and Y is rational. `` developing a of! As we did in the form of a statement in the form of conditional statements in part 4. `` Bonzo is at the moves '' B '' ( A=elephant, B=forgetting ) explanations in last.: write the converse of statement ( 1a ) \wedge \urcorner Q\ ) is logically equivalent if they the... Sometimes referred to as De logically equivalent examples ’ s Laws ( Theorem 2.5 ) to rewrite the of! Are true and which are very complicated from a to B in … equivalence! The given statements in words two important conditional statements equivalence is a tautology is tautology... Any example of two logically equivalent precisely when their truth tables to construct tables for more complicated....

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