logical equivalence statements

The idea is that if \(P \to Q\) is false, then its negation must be true. Definition 2.1.2.. An expression involving logical variables that is false for all values is called a contradiction.. Logical Equivalence : Logical equivalence can be defined as a relationship between two statements/sentences. These statements have the same truth value: If not everyone is happy, then someone is unhappy and vice versa. This is the currently selected item. Its negation is not a conditional statement. Section 2.1 Logical Equivalences Definition 2.1.1.. An expression involving logical variables that is true for all values is called a tautology.. if(vidDefer[i].getAttribute('data-src')) { 1 The conditional statement p !q is logically equivalent to:p_q. If A and B represent statements, then A B means "A if and only if B." For another example, consider the following conditional statement: If \(-5 < -3\), then \((-5)^2 < (-3)^2\). And the easiest way to show equivalence is to create a truth table and see if the columns are identical, as the example below nicely demonstrates. The statement ⌝ ( P ∨ Q) is logically equivalent to ⌝ P ∧ ⌝ Q. Then use one of De Morgan’s Laws (Theorem 2.5) to rewrite the hypothesis of this conditional statement. 1 Logical equivalence When proving a proposition in mathematics it is often useful to look at a logical variation of the proposition in question that \means the same thing". It is asking which statements are logically equivalent to the given statement. The first equivalency in Theorem 2.5 was established in Preview Activity \(\PageIndex{1}\). A compound proposition that is always false is called a contradiction or absurdity. Do not leave a negation as a prefix of a statement. Propositional Logic Grinshpan Examples of logically equivalent statements Here are some pairs of logical equivalences. ... How can logical equivalence be derived from this.. 0. Now, write a true statement in symbolic form that is a conjunction and involves \(P\) and \(Q\). Therefore, an equivalent statement would be of the form. Note: This is not asking which statements are true and which are false. \(\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q\). Preview Activity \(\PageIndex{1}\): Logically Equivalent Statements. That means that a contradiction is when a column is mixed with trues and falses. Label each of the following statements as true or false. Let a and b be integers. Showing logical equivalence or inequivalence is easy. The negation of a conditional statement can be written in the form of a conjunction. The conditional statement \(P \to Q\) is logically equivalent to its contrapositive \(\urcorner Q \to \urcorner P\). The two statements in this activity are logically equivalent. Showing logical equivalence or inequivalence is easy. Then determine which two are logically equivalent. One way of proving that two propositions are logically equivalent is to use a truth table. Now, consider the following statement: If Ryan gets a pay raise, then he will take Allison to dinner. Basically, this means these statements are equivalent, and we make the following definition: 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. 1. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. This means we can also say that If Ryan does not take Allison to dinner, then he did not get a pay raise is logically equivalent. Proof. Logically Equivalent Statement. It is possible to develop and state several different logical equivalencies at this time. Although it is possible to use truth tables to show that \(P \to (Q \vee R)\) is logically equivalent to \(P \wedge \urcorner Q) \to R\), we instead use previously proven logical equivalencies to prove this logical equivalency. For the following, the variable x represents a real number. Propositional Logic Grinshpan Examples of logically equivalent statements Here are some pairs of logical equivalences. The pair of statements cited above illustrate this general fact: Google Classroom Facebook Twitter. Furthermore, there are times when we would instead state reasons for why two statements are logically equivalent, rather than constructing a truth table. Logical equivalence is denoted by this symbol: ≡ Referring back to examples 1.4.1 #4 and #5 we saw that the statement "Some cats are mammals" was true, while the statement "Some cats aren't mammals" was false. And it will be our job to verify that statements, such as p and q, are logically equivalent. Another way to say this is: For each assignment of truth values to the simple statements which make up X and Y, the statements X and Y have identical truth values. Logical Equivalence Recall: Two statements are logically equivalent if they have the same truth values for every possible interpretation. \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\), Biconditional Statement \((P leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)\), Double Negation \(\urcorner (\urcorner P) \equiv P\), Distributive Laws \(P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)\) Table 2.3 establishes the second equivalency. A pet is an animal you own. For example. Conditional and Biconditional Statements. However, in some cases, it is possible to prove an equivalent statement. To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”. \(\urcorner (P \to Q)\) is logically equivalent to \(\urcorner (\urcorner P \vee Q)\). MathJax reference. Consequently, \(p\equiv q\) is same as saying \(p\Leftrightarrow q\) is a tautology. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "license:ccbyncsa", "showtoc:no", "De Morgan\'s Laws", "authorname:tsundstrom2" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FMathematical_Logic_and_Proof%2FBook%253A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)%2F2%253A_Logical_Reasoning%2F2.2%253A_Logically_Equivalent_Statements, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), ScholarWorks @Grand Valley State University, Logical Equivalencies Related to Conditional Statements, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Now, in accordance with the rules and definitions prescribed in logic, we have a plethora of logical equivalences. 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. Logical Equivalences … Does this make sense? Being able to change one proposition for another and maintain its truth value is extremely important. Logical equivalence is different from material equivalence. Knowing that the statements are equivalent tells us that if we prove one, then we have also proven the other. Are the expressions \(\urcorner (P \wedge Q)\) and \(\urcorner P \vee \urcorner Q\) logically equivalent? We notice that we can write this statement in the following symbolic form: \(P \to (Q \vee R)\), The Logic of "If" vs. "Only if" A quick guide to conditional logic… Consider the following conditional statement: Let \(x\) be a real number. What does \meaning the same thing" mean? Suppose we are trying to prove the following: Write the converse and contrapositive of each of the following conditional statements. The relation translates verbally into "if and only if" and is symbolized by a double-lined, double arrow pointing to the left and right (). \(P \to Q\) is logically equivalent to \(\urcorner P \vee Q\). Conditional reasoning and logical equivalence. 1. Because tautologies and contradictions are essential in proving or verifying mathematical arguments, they help us to explain propositional equivalences — statements that are equal in logical argument. Take Calcworkshop for a spin with our FREE limits course. Each may be veri ed via a truth table. It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. The logical equivalence of statement forms P and Q is denoted by writing P Q. var vidDefer = document.getElementsByTagName('iframe'); vidDefer[i].setAttribute('src',vidDefer[i].getAttribute('data-src')); Use previously proven logical equivalencies to prove each of the following logical equivalencies: Conditional Statement. For our purposes, in keeping with our \meaning is truth, truth meaning" mantra, it will mean having the same truth-conditions. Use MathJax to format equations. (e) \(a\) does not divide \(bc\) or \(a\) divides \(b\) or \(a\) divides \(c\). Do not delete this text first. function init() { Justify your conclusion. \(P \to Q \equiv \urcorner Q \to \urcorner P\) (contrapositive) } } } Logical equivalence is denoted by this symbol: ≡ Referring back to examples 1.4.1 #4 and #5 we saw that the statement "Some cats are mammals" was true, while the statement "Some cats aren't mammals" was false. We now define two important conditional statements that are associated with a given conditional statement. Material equivalence is associated with the biconditional. Two forms are equivalent if and only if they have the same truth values, so we con-struct a table for each and compare the truth values (the last column). Another way to say this is: For each assignment of truth values to the simple statementswhich make up X and Y, the statements X and Y have identical truth values. Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? Start studying Logical Equivalences involving conditional statements. As we will see, it is often difficult to construct a direct proof for a conditional statement of the form \(P \to (Q \vee R)\). Preview Activity \(\PageIndex{2}\): Converse and Contrapositive. What do you observe? Learn vocabulary, terms, and more with flashcards, games, and other study tools. If each of the statements can be proved from the other, then it is an equivalent. TABLE 8Logical Equivalences Involving Biconditional Statements. Suppose that the statement “I will play golf and I will mow the lawn” is false. If X, then Y | Sufficiency and necessity. Legal. (f) \(f\) is differentiable at \(x = a\) or \(f\) is not continuous at \(x = a\). If \(P\) and \(Q\) are statements, is the statement \((P \vee Q) \wedge \urcorner (P \wedge Q)\) logically equivalent to the statement \((P \wedge \urcorner Q) \vee (Q \wedge \urcorner P)\)? The Logic of "If" vs. "Only if" A quick guide to conditional logic… Logical equivalencies can be used to simplify statement forms, to confirm or disprove an equivalency, to create efficient and logically correct computer programs, or to aid in the design of digital logic circuits. This means that those two statements are NOT equivalent. Logical Equivalence Laws. This means that those two statements are NOT equivalent. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. Then its negation is true. Others will be established in the exercises. \(P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)\), Conditionals withDisjunctions \(P \to (Q \vee R) \equiv (P \wedge \urcorner Q) \to R\) Okay, so let’s put some of these laws into practice. "It is not Tuesday or it is raining." If we prove one, we prove the other, or if we show one is false, the other is also false. Rosen, Discrete Mathematics and Its Applications, Seventh Edition, p. 28, McGraw-Hill, 2012. This is the currently selected item. So the negation of this can be written as. This can be written as \(\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q\). Since many mathematical statements are written in the form of conditional statements, logical equivalencies related to conditional statements are quite important. (b) Use the result from Part (13a) to explain why the given statement is logically equivalent to the following statement: The second statement is Theorem 1.8, which was proven in Section 1.2. Table 2.3 establishes the second equivalency. (whenever you see $$ ν $$ read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p $$ ν$$ q. Start with. In this case, what is the truth value of \(P\) and what is the truth value of \(Q\)? Conditional reasoning and logical equivalence. If \(x\) is odd and \(y\) is odd, then \(x \cdot y\) is odd. A Dog is an animal. The statement \(\urcorner (P \to Q)\) is logically equivalent to \(P \wedge \urcorner Q\). Have questions or comments? The symbol for this is $$ ν $$ . where \(P\) is“\(x \cdot y\) is even,” \(Q\) is“\(x\) is even,”and \(R\) is “\(y\) is even.” So, the negation can be written as follows: \(5 < 3\) and \(\urcorner ((-5)^2 < (-3)^2)\). We now have the choice of proving either of these statements. Email. This suggests there might be a sort of “algebra” you could apply to statements (okay, there is: it is called Boolean algebra) to transform one statement into another. In Section 2.1, we constructed a truth table for \((P \wedge \urcorner Q) \to R\). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. In doing so, we transform the left-hand side of the statement to match the right-hand side, and we provide reasons for each transformation, similar to constructing a two-column proof in geometry. Okay, so a tautology, usually denoted by a bold-faced capital T, is when an entire column is all true as noted by Oak Ridge National Laboratory. But logical equivalence is much stronger than just having the same truth value. In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. Start studying Logical Equivalences involving conditional statements. Logical Equivalence : Logical equivalence can be defined as a relationship between two statements/sentences. (f) If \(a\) divides \(bc\) and \(a\) does not divide \(c\), then \(a\) divides \(b\). Two statements are Logically Equivalent if they have the same truth table. In propositional logic, logical equivalence is defined in terms of propositional variables: two compound propositions are logically equivalent if they have the same truth values for all possible truth values of the propositional variables they contain. Complete truth tables for \(\urcorner (P \wedge Q)\) and \(\urcorner P \vee \urcorner Q\). ~q p. 2. The pair of statements cited above illustrate this general fact: Get access to all the courses and over 450 HD videos with your subscription, Not yet ready to subscribe? Progress Check 2.7 (Working with a logical equivalency). Write the negation of this statement in the form of a disjunction. The logical equivalency in Progress Check 2.7 gives us another way to attempt to prove a statement of the form \(P \to (Q \vee R)\). (a) \([\urcorner P \to (Q \wedge \urcorner Q)] \equiv P\). Conditional reasoning and logical equivalence. The advantage of the equivalent form, \(P \wedge \urcorner Q) \to R\), is that we have an additional assumption, \(\urcorner Q\), in the hypothesis. Which is the contrapositive of Statement (1a)? This gives us more information with which to work. \(P \to Q \equiv \urcorner P \vee Q\) // Last Updated: January 10, 2021 - Watch Video //. Did you know that the construction of mathematical arguments using compound propositions with the same truth value is used extensively in mathematics and forms the basis for logical equivalence? From a practical point of view, you can replace a statement in a proof by any logically equivalent statement. (c) \(a\) divides \(bc\), \(a\) does not divide \(b\), and \(a\) does not divide \(c\). If Chris doesn't own a pet then he doesn't own a dog. This means that \(\urcorner (P \to Q)\) is logically equivalent to\(P \wedge \urcorner Q\).

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