It is represented by and PÂ Q means "P if and only if Q." These logic proofs can be tricky at first, and will be discussed in much more detail in our “proofs” unit. Use MathJax to format equations. Beside distributive and De Morgan’s laws, remember these two equivalences as well; they are very helpful when dealing with implications. Instead of using truth tables, try to use already established logical equivalencies to justify your conclusions. (a) Write the symbolic form of the contrapositive of \(P \to (Q \vee R)\). \(P \to Q \equiv \urcorner Q \to \urcorner P\) (contrapositive) (f) \(f\) is differentiable at \(x = a\) or \(f\) is not continuous at \(x = a\). The notation is used to denote that and are logically equivalent. Material equivalence is associated with the biconditional. (a) If \(a\) divides \(b\) or \(a\) divides \(c\), then \(a\) divides \(bc\). The negation can be written in the form of a conjunction by using the logical equivalency \(\urcorner (P \to Q) \equiv P \wedge \urcorner Q\). The logical equivalency in Progress Check 2.7 gives us another way to attempt to prove a statement of the form \(P \to (Q \vee R)\). Table 2.3 establishes the second equivalency. \(\urcorner (P \to Q)\) is logically equivalent to \(\urcorner (\urcorner P \vee Q)\). This can be written as \(\urcorner (P \vee Q) \equiv \urcorner P \wedge \urcorner Q\). We notice that we can write this statement in the following symbolic form: \(P \to (Q \vee R)\), Case 1: “ If p then q ” has three equivalent statements. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Formulas p and q are logically equivalent if and only if the statement of their material equivalence (P Q) is a tautology. (e) \(a\) does not divide \(bc\) or \(a\) divides \(b\) or \(a\) divides \(c\). \(P \to Q \equiv \urcorner P \vee Q\) vidDefer[i].setAttribute('src',vidDefer[i].getAttribute('data-src')); Logical equivalence is different from material equivalence. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. It is asking which statements are logically equivalent to the given statement. The two statements in this activity are logically equivalent. Click here to let us know! (b) If \(f\) is not differentiable at \(x = a\), then \(f\) is not continuous at \(x = a\). The following truth table will help to make sense of this. (f) If \(a\) divides \(bc\) and \(a\) does not divide \(c\), then \(a\) divides \(b\). Statements that are not tautologies or contradictions are called contingencies.. Are the expressions logically equivalent? That means that a contradiction is when a column is mixed with trues and falses. Have questions or comments? You do not clean your room and you can watch TV. 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. Justify your conclusion. To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. Which of the following statements have the same meaning as this conditional statement and which ones are negations of this conditional statement? Table 2.3: Truth Table for One of De Morgan’s Laws. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "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 [email protected], status page at https://status.libretexts.org.

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