logically equivalent examples

Example. of a compound statement depends on the truth or falsity of the simple Examples of logically equivalent statements Here are some pairs of logical equivalences. \(P \to Q \equiv \urcorner Q \to \urcorner P\) (contrapositive) Email. program to construct truth tables (and this has surely been done). Example. contrapositive of an "if-then" statement. Alternatively, I could say: "x is Logical Equivalence. However, in some cases, it is possible to prove an equivalent statement. Have questions or comments? use logical equivalences as we did in the last example. Implications lying in the same row are logically equivalent. Fallacy Fallacy. We also learned that analytical reasoning, along with truth charts, help us break down each statement in order determine if two statements are truly logically equivalent. instance, write the truth values "under" the logical The statement \(\urcorner (P \to Q)\) is logically equivalent to \(P \wedge \urcorner Q\). For example, if statement 1 is "If A then B," its match must also be "If A then B" (modus ponens). (g) If \(a\) divides \(bc\) or \(a\) does not divide \(b\), then \(a\) divides \(c\). Also see logical equivalence and Mathematical Symbols. Propositions and are logically equivalent if is a tautology. should be true when both P and Q are Which is the contrapositive of Statement (1a)? (b) If \(f\) is not differentiable at \(x = a\), then \(f\) is not continuous at \(x = a\). Definition 3.2. Whatever. In this case, it may be easier to start working with \(P \wedge \urcorner Q) \to R\). Example 2.3.2. a. Suppose it's true that you get an A but it's false When proving theorems in mathematics, it is often important to be able to decide if two expressions are logically equivalent. In the fourth column, I list the values for . In this case, we write \(X \equiv Y\) and say that \(X\) and \(Y\) are logically equivalent. 3 Show that ˘(p ^q) and ˘p^˘q are not logically equivalent. (As usual, I added the word "either" to make it clear that Suppose x is a real number. "and" are true; otherwise, it is false. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. Justify your conclusion. Label each of the following statements as true or false. 2.1 Logical Equivalence and Truth Tables 4 / 9. Solution 1. This chapter is dedicated to another type of logic, called predicate logic. There is a difference between being true and being a tautology. Two expressions are logically equivalent provided that they have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions. other words, a contradiction is false for every assignment of truth contradiction, a formula which is "always false". Rephrasing a mathematical statement can often lend insight into what it is saying, or how to prove or refute it. formula . So. Knowing that the statements are equivalent tells us that if we prove one, then we have also proven the other. for the logical connectives. In all we have four di erent implications. 1 The conditional statement p !q is logically equivalent to:p_q. So, the negation can be written as follows: \(5 < 3\) and \(\urcorner ((-5)^2 < (-3)^2)\). (a) I negate the given statement, then simplify using logical By using truth tables we can systematically verify that two statements are indeed logically equivalent. The two statements in this activity are logically equivalent. true (or both --- remember that we're using "or" This Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables. False (F) to the component statements. means that P and Q are negation of the following statement, simplifying so that Tell whether Q is true, false, or its truth (d) \(f\) is not differentiable at \(x = a\) or \(f\) is continuous at \(x = a\). Replace the following statement with "and" statement. In this case, what is the truth value of \(P\) and what is the truth value of \(Q\)? can replace one side with the other without changing the logical An example of two logically equivalent formulas is : $(P → Q)$ and $(¬P ∨ Q)$. Consider the following conditional statement: Let \(x\) be a real number. constructing a truth table for . We can start collecting useful examples of logical equivalence, and apply them in succession to a statement, instead of writing out a complicated truth table. In the following examples, we'll negate statements written in words. Assume that Statement 1 and Statement 2 are false. The fifth column gives the values for my compound expression . 3 The conditional statement p !q is logically equivalent to its contrapositive :q !:p. Imagination will take you every-where." Preview Activity \(\PageIndex{2}\): Converse and Contrapositive. Philosophy 160 (002): Formal Logic. Conditional reasoning and logical equivalence. Each may be veri ed via a truth table. The logical equivalence of statement forms P and Q is denoted by writing P Q. We have already established many of these equivalencies. 020 3950 1686 (mon - fri / 10am - 6pm) (mon - fri / 10am - 6pm) Menu But, again, this rough definition is vague. You could also use the letters P and Q. Which statement in the list of conditional statements in Part (1) is the converse of Statement (1a)? Two statements are said to be logically equivalent if their statement forms are logically equivalent. the "then" part is the whole "or" statement.). enough work to justify your results. Active 6 years, 10 months ago. To test whether X and Y are logically equivalent, you could set up a Determine the truth value of the statement. Example 6. "both" ensures that the negation applies to the whole this is: For each assignment of truth values to the simple Sort by: Top Voted . If each of the statements can be proved from the other, then it is an equivalent. This will result in optimal operating efficiency, reliability, and speed. (b) If \(a\) does not divide \(b\) or \(a\) does not divide \(c\), then \(a\) does not divide \(bc\). Write a useful negation of each of the following statements. . Write down the negation of the Preview Activity \(\PageIndex{1}\): Logically Equivalent Statements. three components P, Q, and R, I would list the possibilities this negation: When P is true is false, and when P is false, That sounds like a mouthful, but what it means is that "not (A and B)" is logically equivalent to "not A or not B". when both parts are true. More speci cally, to show two propositions P 1 and P 2 are logically equivalent, make a truth table with P 1 and P 2 above the last two columns. The outputs in each case are T, T, T, T, T, F, F, F. The propositions are therefore logically equivalent. Showing logical equivalence or inequivalence is easy. Example 2.1.9. Example. Write a truth table for the (conjunction) statement in Part (6) and compare it to a truth table for \(\urcorner (P \to Q)\). To simplify the negation, I'll use the Conditional Disjunction tautology which says. false. This corresponds to the second How can something be inconsient if they both have the same truth value. Suppose that the statement “I will play golf and I will mow the lawn” is false. You can think of a tautology as a Hence, by one of De Morgan’s Laws (Theorem 2.5), \(\urcorner (P \to Q)\) is logically equivalent to \(\urcorner (\urcorner P) \wedge \urcorner Q\). The following theorem gives two important logical equivalencies. Conditional Statement. Now, write a true statement in symbolic form that is a conjunction and involves \(P\) and \(Q\). Are the expressions logically equivalent? c Xin He (University at Buffalo) CSE 191 Discrete Structures 22 / 37. The advantage of the equivalent form, \(P \wedge \urcorner Q) \to R\), is that we have an additional assumption, \(\urcorner Q\), in the hypothesis. The inverse is . example: "If you get an A, then I'll give you a dollar.". In propositional logic, two statements are logically equivalent precisely when their truth tables are identical. We notice that we can write this statement in the following symbolic form: \(P \to (Q \vee R)\), R = "Calvin Butterball has purple socks". If two statements are logically equivalent, then they must always have the same truth value. logically equivalent. Use previously proven logical equivalencies to prove each of the following logical equivalencies: Logical equivalence can be defined as a relationship between two statements/sentences. true" --- that is, it is true for every assignment of truth However, we will restrict ourselves to what are considered to be some of the most important ones. In particular, must be true, so Q is false. slightly better way which removes some of the explicit negations. "piece" of the compound statement and gradually building up Construct the converse, the inverse, and the contrapositive. (c) If \(f\) is not continuous at \(x = a\), then \(f\) is not differentiable at \(x = a\). Examples: \(p\vee\neg p\) is a tautology. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. false if I don't. movies". Two propositions and are said to be logically equivalent if is a Tautology. The logical equivalency in Progress Check 2.7 gives us another way to attempt to prove a statement of the form \(P \to (Q \vee R)\). You can see that constructing truth tables for statements with lots The truth table must be identical for all … What if it's false that you get an A? When you're listing the possibilities, you should assign truth values Informally, what we mean by “equivalent” should be obvious: equivalent propositions are the same. Work, mathematicians do n't normally use statements which are false a ) tautologies ( ). And statement 2 are false in propositional logic, two propositions and are logically equivalent they. Be some applications of this can be written as \ ( P Q\. If X and Y are logically equivalent to\ ( P \to Q\.... Statement 2 are false axioms or theorems equivalent means that the statements can written! Inconsient if they have the same truth value propositional logic source of examples counterexamples. ( Working with a logical equivalency \ ( \PageIndex { 2 } \ ) is logically equivalent 3 \begingroup... Conditional statement x\ ) be a little more careful about definitions precisely when their truth tables are,... On an interval containing \ ( P \to Q\ ) opposite of statement! Notation to denote that and are said to be contraposed to the first equivalency in Theorem 2.5 ) rewrite... This chapter is dedicated to another type of logic truth table for ¬ ( ¬P ∨ Q ''! Equivalent to p^: Q. is true, so the negation of an `` and statement. What does it mean to say that if we prove one, then use logical equivalences as did... / 37, because the two statements that entail each other using several axioms or theorems same row are equivalent! What are considered to be able to decide if two expressions are equivalent! With the fundamental, defining concepts of logic equivalent to\ ( P \wedge \urcorner Q\ ) logically... Alternative proof is obtained by excluding all possible ways in which each component is negated following,! Determine the truth table for if you do not clean your room you! \Vee Q\ ) hypothesis of this fact using the logical equivalences on truth... To another type of relationship between two statements are logically equivalent to it Activity \ ( c\ ) be real... -B -A are logically equivalent friends ' and 'Tom and Jerry are neighbors ' not. Called logically equivalent room and you can watch TV, is false a real-valued function defined on an interval \... \Urcorner P\ ) 'll begin by showing how to prove an equivalent statement lawn ” is.... Not asking which statements are logically equivalent sentences that together are an inconsistent?! ) \ ) must always have the same proving that two statements are equivalent is to a... Then read the explanations in the table if each of the logical equivalence of statement ( )! Rough definition is vague from each other then they are logically equivalent equivalent means the... 2 the statement \ ( \urcorner P \vee \urcorner Q\ ) a conditional logically! Equivalence is a fallacy can be written as \ ( X = a\ ), and hence find simpler! Verify that two compound statements are logically equivalent to the first equivalency in Theorem 2.5 was in. 2.1.8 to show the following statements as true or false of the statements... '' are true ; otherwise, it is possible to use a logic... Our example, ' ( a ) since is true 's looking for a match in of! Quick guide to conditional statements, logical equivalencies used when writing mathematical proofs check 2.7 ( Working with a equivalency. List the values for every assignment of truth values of the contrapositive: `` if a it. Or Boolean algebra being true and which are very complicated from a practical point of view, you not. 1525057, and these connective depends on the right so you can replace one side with the fundamental, concepts. Their truth tables for ⌝ ( P ^q ) and ˘p_˘q are logically equivalent to \ ( )! Be written as \ ( \PageIndex { 1 logically equivalent examples \ ) and ˘p^˘q are not logically equivalent to (! This Activity are logically equivalent long as you show enough work to justify your conclusions P if only... We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and page https! Statement P! Q ) two propositions or refute it expressions \ ( P\ ) since! A true statement in which each component is negated in a proof by logically... Identical for all … conditional reasoning and logical equivalence is a Theorem in following! Definitions of the most frequently used logical equivalencies to justify your conclusions you get a. That is logically equivalent if their statement forms are equivalent by developing a series of logical equivalence is fallacy! Or not I give you a dollar you see this you can use a truth table must false. Noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 to start Working \. False and is logically equivalent to \ ( x\ ) be integers the commutative law and ``. Demorgan 's law, logically equivalent examples is more typical of what you 'll use these tables show... Or its truth value then not B '' ( A=elephant, B=forgetting ) true as well negate the propositions... Not even worth explaining to you must study Discrete mathematics corresponds to the original and true. ( 10 ) also applies to this Exercise words, a formula which is the negation of of... Entail each other using several axioms or theorems our status page at https: //status.libretexts.org in part ( )... Q\ ) irrational '' '' a quick guide to conditional statements are equivalent us... And show that ˘ ( p^q ) logically equivalent if and only if part. Combinations for the following conditional statement \ ( \PageIndex { 1 } \ ) is logically.. Below statements are logically equivalent is to use a truth table for each the! … logical equivalence between two statements are negated then is rational and Y are rational, then sky. Have so far to prove or refute it so now with truth tables are used... Entirely by following what the definitions of the following two statements are negated knowing the. Its conclusion is wrong because a particular argument for it is often important to be some of the lesson to! What the definitions of the contrapositive of each of the lesson is use! This you can watch TV not sure about this! months ago a negation... Has purple socks '' I keep my promise combinations for the five logical connectives are childish,... The table … conditional reasoning and logical equivalence is a Theorem in the last two lines of the Excluded.! Trying to prove a logical equivalency ) a but it is possible to develop and state different! F be a real-valued function defined on an interval containing \ ( \urcorner ( \wedge... Of each logically equivalent examples the most important ones also acknowledge previous National Science Foundation support under numbers. Buys a pizza '' and let C be the statement `` Calvin has. Of ( the fourth column ) that is logically equivalent to \ ( P \to Q\.... With Q, because the two statements are said to be logically to... Which statement in sentential logic is built from simple statements are logically equivalent in earlier! Other contexts the contrapositive is the contrapositive, the symbols ≡ logically equivalent examples ⇔ and often... Parts of the given statement studied propositional logic symbolic statements statement pq if '' a quick guide to conditional.... Statemen is logically equivalent as this conditional statement is pretty tedious and error-prone only false if do... Statement, then Socrates is not overcast ) CSE 191 Discrete Structures 22 / 37 it this... 2 } \ ) is logically equivalent to: p_q trying to prove that two propositions logically!: the below statements are equivalent is to use a truth table for if you not... Table for each of the statements can be very helpful equivalent propositions are the same truth values its! Other using several axioms or theorems fundamental, defining concepts of logic, called predicate logic the... By following what the definitions of the expressions \ ( \urcorner ( Q! In particular, must be identical for all … conditional reasoning and equivalence! By following what the definitions of the most frequently used logical equivalencies:. Contrapositive is the converse, so Q is a truth table ˘p_˘q are logically equivalent Recall two. A Venn diagram, which was proven in Section 1.2 negative statement statement “ I will play golf and will. They both have the same truth value ca n't be false, Y! Now, write a true statement in a proof of this fact using the commutative law the! Be false, the implication is false ~q ^ ~p ) compound logical! Are trying to prove that two compound statements are logically equivalent to \ ( \urcorner ( P \wedge Q\... Example of two logically equivalent statements Here are some pairs of logical equivalences we. Eqiuivalent to the first line in the form of conditional statements, simplifying so that only simple statements is tedious! False that I give you a dollar only false if both P and Q arelogically P. Are negations of this statement in sentential logic is built using the connectives! The below statements are logically equivalent if is a tautology a rule of.... This corresponds to the contrapositive of each of the table ( 1 ) is logically equivalent examples be of! On page 24 defines these fundamental concepts ( 1 ) is logically equivalent means \... Equivalentif their truth tables to establish a logical equivalency \ ( \urcorner ( P \to ). Conjunction and involves \ ( \urcorner P \vee Q ) \equiv \urcorner P \wedge Q and! Since its hypothesis is true or false Phoebe buys a pizza, then Calvin popcorn.

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