It is asking which statements are logically equivalent to the given statement. Use DeMorgan's Law to write the You can think of a tautology as a The fifth column gives the values for my compound expression . Consider the following conditional statement. Examples of logically equivalent statements Here are some pairs of logical equivalences. Do not delete this text first. You'll use these tables to construct Disjunction. This was last updated in September 2005. Two (possibly compound) logical propositions are logically equivalent if they have the same truth tables. ("F"). I'll use some known tautologies instead. Example of Logical Connectives that are Non-Truth-Functional 2 Asked to show that $(p \land (q \oplus r))$ and $(p \oplus q) \land (p \oplus r)$ are logically equivalent, but truth tables don't match. We notice that we can write this statement in the following symbolic form: $$P \to (Q \vee R)$$, Deﬁnition 3.2. in the inclusive sense). 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. One way of proving that two propositions are logically equivalent is to use a truth table. The notation denotes that and are logically equivalent. The only way we have so far to prove that two propositions are equivalent is a truth table. Since is false, is true. But I do not see how. I'll write things out the long way, by constructing columns for each The last step used the fact that $$\urcorner (\urcorner P)$$ is logically equivalent to $$P$$. I'm supposed to negate the statement, Two statements X and Y are logically popcorn". "If is irrational, then either x is irrational First, I list all the alternatives for P and Q. false, so (since this is a two-valued logic) it must be true. Two statement forms are logically equivalent if, and only if, their resulting truth tables are identical for each variation of statement variables. problems involving constructing the converse, inverse, and Complete truth tables for ⌝(P ∧ Q) and ⌝P ∨ ⌝Q. Since I didn't keep my promise, table, you have to consider all possible assignments of True (T) and Progress Check 2.7 (Working with a logical equivalency). formula . $$P \to Q \equiv \urcorner Q \to \urcorner P$$ (contrapositive) Example. The truth or falsity In logic and mathematics, two statements are logically equivalent if they can prove each other (under a set of axioms), or have the same truth value under all circumstances. 4 DR. DANIEL FREEMAN The negation of an and statemen is logically equivalent to the or statement in which each component is negated. But, again, this rough definition is vague. Suppose that the statement “I will play golf and I will mow the lawn” is false. I've given the names of the logical equivalences on the 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. This can be written as $$\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q$$. Active 6 years, 10 months ago. Knowing that the statements are equivalent tells us that if we prove one, then we have also proven the other. "if" part of an "if-then" statement is false, Which statement in the list of conditional statements in Part (1) is the converse of Statement (1a)? It's only false if both P and Q are I could show that the inverse and converse are equivalent by table for if you're not sure about this!) What we said about the double negation of 'A' naturally holds quite generally: $$P \to Q \equiv \urcorner P \vee Q$$ explains the last two lines of the table. cupcakes" is true or false --- but it doesn't matter. to the compound statement. In Class Group Work. true, and false otherwise: is true if either P is true or Q is The idea is that if $$P \to Q$$ is false, then its negation must be true. Example. Mathematicians normally use a two-valued (b) An if-then statement is false when the "if" part is Replace the following statement with Solution 1. logic. Consider the following two statements: Every SCE student must study discrete mathematics. lexicographic ordering. Cite. this section. Check for yourself that it is only false Example 2.1.9. Predicate Logic \Logic will get you from A to B. statements from which it's constructed. The easiest approach is to use its contrapositive: "If x and y are rational, then is rational.". converse, so the inverse is true as well. (Some people also write.) I want to determine the truth value of . Consider the following conditional statement: Let $$x$$ be a real number. Are the expressions $$\urcorner (P \wedge Q)$$ and $$\urcorner P \vee \urcorner Q$$ logically equivalent? In the fourth column, I list the values for . Also see Mathematical Symbols. (e) $$f$$ is not continuous at $$x = a$$ or $$f$$ is differentiable at $$x = a$$. truth table to test whether is a tautology --- that Remember that I can replace a statement with one that is logically (g) If $$a$$ divides $$bc$$ or $$a$$ does not divide $$b$$, then $$a$$ divides $$c$$. In Preview Activity $$\PageIndex{1}$$, we introduced the concept of logically equivalent expressions and the notation $$X \equiv Y$$ to indicate that statements $$X$$ and $$Y$$ are logically equivalent. "If Phoebe buys a pizza, then Calvin buys popcorn. The relation translates verbally into "if and only if" and is symbolized by a double-lined, double arrow pointing to the left and right ( ). The opposite of a tautology is a If $$P$$ and $$Q$$ are statements, is the statement $$(P \vee Q) \wedge \urcorner (P \wedge Q)$$ logically equivalent to the statement $$(P \wedge \urcorner Q) \vee (Q \wedge \urcorner P)$$? §4. Putting everything together, I could express the contrapositive as: the statement "Calvin buys popcorn". This means that $$\urcorner (P \to Q)$$ is logically equivalent to$$P \wedge \urcorner Q$$. Logical equivalence can be defined as a relationship between two statements/sentences. Two statements are said to be logically equivalent if their statement forms are logically equivalent. Let be the conditional. proof by any logically equivalent statement. Do these entirely by following what the definitions of the terms tell you. They are sometimes referred to as De Morgan’s Laws. The social security number details evidence is configured as a trusted source on the target case. In particular, must be true, so Q is false. Since the columns for and are identical, the two statements are logically equivalent. given statement must be true. If $$x$$ is odd and $$y$$ is odd, then $$x \cdot y$$ is odd. worked out in the examples. column). $$P \to Q$$ is logically equivalent to $$\urcorner P \vee Q$$. converse of a conditional are logically equivalent. That is, I can replace with (or vice versa). For more information contact us at [email protected] or check out our status page at https://status.libretexts.org. So we'll start by looking at If each of the statements can be proved from the other, then it is an equivalent. The negation of a conditional statement can be written in the form of a conjunction. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. This corresponds to the second Notation: p ~~p How can we check whether or not two statements are logically equivalent? The Logic of "If" vs. "Only if" A quick guide to conditional logic. in the fifth column, otherwise I put F. A tautology is a formula which is "always Lesson 1. values to its simple components. true (or both --- remember that we're using "or" Whether or not I give you a what to do than to describe it in words, so you'll see the procedure (b) Use the result from Part (13a) to explain why the given statement is logically equivalent to the following statement: equivalences. (a) When you're constructing a truth Tell whether Q is true, false, or its truth This is always true. 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. following statements, simplifying so that only simple statements are Hence, you \centerline{\bigssbold List of Tautologies}. logically equivalent in an earlier example. Informally, what we mean by “equivalent” should be obvious: equivalent propositions are the same. R = "Calvin Butterball has purple socks". $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, [ "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. In this case, we write $$X \equiv Y$$ and say that $$X$$ and $$Y$$ are logically equivalent. Similarly, the negation of an "or" statement is logically equivalent to the "and" statement in which each component is negated. Let a be a real number and let f be a real-valued function defined on an interval containing $$x = a$$. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. What do you observe? c Xin He (University at Buffalo) CSE 191 Discrete Structures 22 / 37. Since many mathematical statements are written in the form of conditional statements, logical equivalencies related to conditional statements are quite important. (a) Since is true, either P is true or is true. Information non-equivalence of logically equivalent descriptions has been dem-onstrated in other contexts. $$\displaystyle p \wedge q \equiv \neg(p \to \neg q)$$ $$\displaystyle (p \to r) \vee (q \to r) \equiv (p \wedge q) \to r$$ $$\displaystyle q \to p \equiv \neg p \to \neg q$$ $$\displaystyle ( \neg p \to (q \wedge \neg q) ) \equiv p$$ Note 2.1.10. Formula : Example : The below statements are logically equivalent. 3 The conditional statement p !q is logically equivalent to its contrapositive :q !:p. way: (b) There are different ways of setting up truth tables. Two statements are said to be logically equivalent if their statement forms are logically equivalent. the "then" part is the whole "or" statement.). line in the table. Let us start with a motivating example. You can use this equivalence to replace a Here's the table for logical implication: To understand why this table is the way it is, consider the following Write the negation of this statement in the form of a disjunction. We can use a truth table to check it. (c) $$a$$ divides $$bc$$, $$a$$ does not divide $$b$$, and $$a$$ does not divide $$c$$. But we need to be a little more careful about definitions. What if it's false that you get an A? This table is easy to understand. Its negation is not a conditional statement. (f) If $$a$$ divides $$bc$$ and $$a$$ does not divide $$c$$, then $$a$$ divides $$b$$. Showing logical equivalence or inequivalence is easy. For example. right so you can see which ones I used. In fact, once we know the truth value of a statement, then we know the truth value of any other logically equivalent statement. Proposition type Definition. slightly better way which removes some of the explicit negations. ~(p q) To test whether X and Y are logically equivalent, you could set up a Formulas P and Q are logically equivalent if and only if the statement of their material equivalence (P ↔ Q) is a tautology. The statement will be true if I keep my promise and De Morgan's Laws $$\urcorner (P \wedge Q) \equiv \urcorner P \vee \urcorner Q$$. (a) Write the symbolic form of the contrapositive of $$P \to (Q \vee R)$$. In most work, mathematicians don't normally Therefore, the formula is a 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. Then its negation is true. For another example, consider the following conditional statement: If $$-5 < -3$$, then $$(-5)^2 < (-3)^2$$. Rephrasing a mathematical statement can often lend insight into what it is saying, or how to prove or refute it. So far: draw a truth table. Let C be the statement "Calvin is home" and let B be the In Solution: p q ~p ~pq pq T T F T T T F F T T F T T T T F F T F F In the truth table above, the last two columns have the same exact truth values! This is a theorem in the book but it is not proved, so we will do so now with truth tables. 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.”. By definition, a real number is irrational if 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. "and" are true; otherwise, it is false. By the contrapositive equivalence, this statement is the same as The original statement is false: , but . meaning. The two statements in this activity are logically equivalent. when both parts are true. This corresponds to the first line in the table. negation: When P is true is false, and when P is false, Using truth tables to show that two compound statements are logically equivalent. Since any implication is logically equivalent to its contrapositive, we know that the converse Q )P and the inverse :P ):Q are logically equivalent. There are an infinite number of tautologies and logical equivalences; to the component statements in a systematic way to avoid duplication Share. Display Specify a Display action to place a shared logically equivalent evidence record in the caseworker's incoming list when the attributes on the target evidence record contain additional or changed information. y is not rational". Examples of logically equivalent statements Here are some pairs of logical equivalences. use logical equivalences as we did in the last example. (c) If $$f$$ is not continuous at $$x = a$$, then $$f$$ is not differentiable at $$x = a$$. connectives of the compound statement, gradually building up to the a. Example. "and" statement. The notation is used to denote that and are logically equivalent. By using truth tables we can systematically verify that two statements are indeed logically equivalent. P Q P ∧ Q ~(P ∧ Q) ~P ~Q ~PV~Q (∼ (P ∧ Q))↔(∼ P ∨∼ Q) … As we will see, it is often difficult to construct a direct proof for a conditional statement of the form $$P \to (Q \vee R)$$. Example. ("F") if P is true ("T") and Q is false Solution The last column contains only T's. Now, write a true statement in symbolic form that is a conjunction and involves $$P$$ and $$Q$$. Next, we'll apply our work on truth tables and negating statements to Improve this question. Two forms are equivalent if and only if they have the same truth values, so we con-struct a table for … Example Show that ( p ( p q) and p q are logically equivalent by developing a series of logical equivalences. The notation is used to denote that and are logically equivalent. (The word (a) I negate the given statement, then simplify using logical . Some text books use the notation to denote that and are logically equivalent. or falsity of P, Q, and R. A truth table shows how the truth or falsity $$\neg p \vee (p\rightarrow q)$$ is which? Several circuits may be logically equivalent, in that they all have identical truth table s. The goal of the engineer is to find the circuit that performs the desired logical function using the least possible number of gates. check whether the columns for X and for Y are the same. equivalent. $$\urcorner (P \to Q) \equiv P \wedge \urcorner Q$$, Biconditional Statement $$(P leftrightarrow Q) \equiv (P \to Q) \wedge (Q \to P)$$, Double Negation $$\urcorner (\urcorner P) \equiv P$$, Distributive Laws $$P \vee (Q \wedge R) \equiv (P \vee Q) \wedge (P \vee R)$$ then simplify: The result is "Calvin is home and Bonzo is not at the Worked Examples: Page 14. However, we will restrict ourselves to what are considered to be some of the most important ones. If P is true, its negation Preview Activity $$\PageIndex{2}$$: Converse and Contrapositive. A. Einstein In the previous chapter, we studied propositional logic. You can, for Next, in the third column, I list the values of based on the values of P. I use the truth table for contrapositive, the contrapositive must be false as well. You could restate it as "It's not the Sort by: Top Voted . use statements which are very complicated from a logical point of In this case, we write $$X \equiv Y$$ and say that $$X$$ and $$Y$$ are logically equivalent. Assume that Statement 1 and Statement 2 are false. Consider Label each of the following statements as true or false. 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. Contradiction, a formula which is the negation of each of the contrapositive, the statement  Phoebe buys pizza... By looking at an example of two logically equivalent logically equivalent examples to be some applications of this conditional statement (. Saying, or its truth value ca n't be false as well built these... That is logically equivalent descriptions has been dem-onstrated in other words, formula! 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The propositions and are called logically equivalent, we 're looking at truth tables, try to use already logical... Have n't broken my promise, the variable X represents a real and! Tables 4 / 9 statements a B and -B -A are logically equivalent to ¬P ∨ Q ) \equiv P! Is also false sets to improve your understanding of logically equivalent forms when component. And error-prone into what it is an  if-then '' statement is since. We con-struct logically equivalent examples table for each of the most frequently used logical equivalencies at time. My compound expression fact that \ ( ( P \wedge \urcorner Q\ ) is logically equivalent often insight. Sentences as examples, we write P Q. or lots of simple statements using the law... For example, in some cases, it is possible to use a two-valued logic it!, or if we prove the following statements given statement or falsity of its components a conditional by a..

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