However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions \(P\text{,}\) \(Q\text{,}\) etc. For all numbers \(n\text{,}\) if \(n\) is prime, then \(n+3\) is not prime. OR (∨) 2. Geoff Poshingten is out at a fancy pizza joint, and decides to order a calzone. Tautologies are always true but they don't tell us much about the world. Notice that the above example illustrates that the negation of an implication is NOT an implication: it is a conjunction! }\), \(\neg(\exists x P(x) \imp \forall y P(y))\text{.}\). In fact, it is equally true that “If the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.”, You might have noticed in Example 3.1.1 that the final column in the truth table for \(\neg P \vee Q\) is identical to the final column in the truth table for \(P \imp Q\text{:}\). Look at the truth value of \(Q\) in each of the rows that have \(P \vee Q\) and \(\neg P\) true. Since the truth value of a statement is completely determined by the truth values of its parts and how they are connected, all you really need to know is the truth tables for each of the logical connectives. }\) The second allows different \(y\)'s to work for different \(x\)'s, but there is nothing preventing us from using the same \(y\) that work for every \(x\text{. \end{equation*}, \begin{equation*} And lo-and-behold, in this one case, \(Q\) is also true. Oh, and if I have pepperoni or quail then I must also have ricotta cheese.”. With this and De Morgan's laws, you can take any statement and simplify it to the point where negations are only being applied to atomic propositions. What did Tommy eat? Everything that we learned about logical equivalence and deductions still applies. }\) If Geoff is a truth-teller, then all three statements would be true. Let \(P\) denote “Edith eats her vegetables” and \(Q\) denote “Edith can have a cookie.” The logical form of the argument is then: This is an example of a deduction rule, an argument form which is always valid. P \imp Q \text{ is logically equivalent to } \neg P \vee Q\text{.} This is sort of like a tautology, although we reserve that term for necessary truths in propositional logic. Let's look at the form of the statements. Nearly all discrete math classes offered by computer science departments include work in propositional logic. Then, the last column is determined by the values in the previous two columns and the definition of \(\vee\text{. }\), \(\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))\text{. \end{equation*}, \begin{equation*} Enter truth tables. Start with \(\neg(P \imp Q)\text{. It would be a good idea to use only conjunctions, disjunctions, and negations. There is a number \(n\) for which no other number is either less \(n\) than or equal to \(n\text{.}\). Determine if the following is a valid deduction rule: Can you chain implications together? }\), \(\neg((\neg P \wedge Q) \vee \neg(R \vee \neg S))\text{.}\). Simplifying negations will be especially useful in the next section when we try to prove a statement by considering what would happen if it were false. Propositional Logic – Wikipedia Principle of Explosion – Wikipedia Discrete Mathematics and its Applications, by Kenneth H Rosen. We saw this before, in Section 0.2, but it is so important and useful, it warants a second blue box here: The negation of an implication is a conjuction: That is, the only way for an implication to be false is for the hypothesis to be true AND the conclusion to be false. Outline 1 Propositions 2 Logical Equivalences 3 Normal Forms Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Then we can clearly see in which cases the statement is true or false. We can translate as follows: In this case, we are using \(P(x)\) to denote “\(x\) is prime” and \(O(x)\) to denote “\(x\) is odd.” These are not propositions, since their truth value depends on the input \(x\text{. True or false? Mathematics | Propositional Equivalences. Don't just say, “it is false that …”. Instead, you should use part (a) and mathematical induction. Consider the three rows that evaluate to false and say what the truth values of \(T\text{,}\) \(S\text{,}\) and \(P\) are there. We cannot do the reverse of this though. Then to fill in the final column, look only at the column for \(Q\) and the column for \(\neg P\) and use the rule for \(\vee\text{.}\). }\) This literally says, “for every number \(x\) there is a number \(y\) which is smaller than \(x\text{. Let's see how we can apply the equivalences we have encountered so far. What else did he wear? \newcommand{\va}[1]{\vtx{above}{#1}} The premises in this case are \(P \imp Q\) and \(P\text{. Yesterday, Holmes wore a bow tie. Master Discrete Mathematics: Sets, Math Logic, and More. Make a truth table for each and compare. This one is a particularly famous rule called modus ponens. }\) In other words, while we don't have logical equivalence between the two statements, we do have a valid deduction rule: Put yet another way, this says that the single statement. This is a course in discrete mathematics; Chocolate cupcakes are the best Make a truth table for the statement \(\neg P \wedge (Q \imp P)\text{. To see this, we should provide an interpretation of the predicate \(P(x,y)\) which makes one of the statements true and the other false. We want to know whether \(\neg(P \vee Q)\) is logically equivalent to \(\neg P \wedge \neg Q\text{. Either Sam is a woman and Chris is a man, or Chris is a woman. That is, if \(P \imp Q\) and \(Q \imp R\text{,}\) does that means the \(P \imp R\text{? Here we talk about all the laws that we are gonna use from now on to solve all the problems. \newcommand{\N}{\mathbb N} \newcommand{\imp}{\rightarrow} \neg(P \wedge Q) \text{ is logically equivalent to } \neg P \vee \neg Q\text{.} Can you switch the order of quantifiers? What can you conclude? Here we go through the first lecture of our curriculum, talking about Propositional Logic. Suppose further that, is a valid deduction rule. There will be three rows in which the statement is false. Translate Geoff's order into logical symbols. Those are true if either \(P\) is false or \(Q\) is true (in the first case) and \(Q\) is false or \(R\) is true (in the second case). Prove that the statements \(\neg(P \imp Q)\) and \(P\wedge \neg Q\) are logically equivalent without using truth tables. \newcommand{\card}[1]{\left| #1 \right|} Which is it? }\) Explain how you know you are correct. If you believed the statement was false, what properties would a counterexample need to possess? }\)” We see that this is another way to make our original claim. Are the statements, “it will not rain or snow” and “it will not rain and it will not snow” logically equivalent? For the second part, you can inductively assume that from the first \(n-2\) implications you can deduce \(P_1 \imp P_{n-1}\text{. Rephrasing a mathematical statement can often lend insight into what it is saying, or how to prove or refute it. If he was a liar, then all three statements would be false. Assuming the statement is true, what (if anything) can you conclude if there will be cake? \((P \wedge Q) \wedge (R \wedge \neg R)\text{. That is, \(P\) and \(Q\) have the same truth value under any assignment of truth values to their atomic parts. If not, consider the following truth table: This is just the truth table for \(P \imp Q\text{,}\) but what matters here is that all the lines in the deduction rule have their own column in the truth table. \newcommand{\gt}{>} Here is the truth table: We added a column for \(\neg P\) to make filling out the last column easier. \newcommand{\C}{\mathbb C} The idea is this: on each row, we list a possible combination of T's and F's (for true and false) for each of the sentential variables, and then mark down whether the statement in question is true or false in that case. Suppose \(P_1, P_2, \ldots, P_n\) and \(Q\) are (possibly molecular) propositional statements. \end{equation*}, \begin{equation*} Could both trolls be knights? Luckily, we can make a chart to keep track of all the possibilities. Recognizing two statements as logically equivalent can be very helpful. 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