Now look at all the rows for which both \(P \imp Q\) and \(\neg P \imp Q\) are true. Simplify the statements below (so negation appears only directly next to predicates). So if \(P\imp Q\) and \(P\) are both true, we see that \(Q\) must be true as well. A proposition is simply a statement. In fact, it is equally true that âIf the moon is made of cheese, then Elvis is still alive, or if Elvis is still alive, then unicorns have 5 legs.â, You might have noticed in ExampleÂ 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 \(P \imp Q\text{:}\). \neg(\neg P \vee Q)\text{.} Don't just say, âit is false that â¦â. }\) The first is saying we can find one \(y\) that works for every \(x\text{. \neg \exists x \forall y (x \le y) Here they are: The truth table for negation looks like this: None of these truth tables should come as a surprise; they are all just restating the definitions of the connectives. Negation/ NOT (¬) 4. You are looking for a row in which \(P\) is true, and the whole statement is true. Note that this statement is not \(\neg(P \vee Q)\text{,}\) the negation belongs to \(P\) alone. To verify that two statements are logically equivalent, you can use truth tables or a sequence of logically equivalent replacements. Rather, we end with a two examples of logical equivalence and deduction, to pique your interest. \(\neg \exists x \forall y (\neg O(x) \vee E(y))\text{. There is a sequence that is both arithmetic and geometric. However, predicate logic allows us to analyze statements at a higher resolution, digging down into the individual propositions \(P\text{,}\) \(Q\text{,}\) etc. Hint: you should get three T's and one F. It's your birthday, but the cake is a lie. Suppose further that, is a valid deduction rule. Edith ate her vegetables. Propositional logic studies the ways statements can interact with each other. \neg \neg P \text{ is logically equivalent to } P\text{.} }\), \(\neg((P \imp \neg Q) \vee \neg (R \wedge \neg R))\text{. Soâyeah, it gets kind of messy. That is, \(P\) and \(Q\) have the same truth value under any assignment of truth values to their atomic parts. \newcommand{\vl}[1]{\vtx{left}{#1}} Now let's answer our question about monopoly: Analyze the statement, âif you get more doubles than any other player you will lose, or that if you lose you must have bought the most properties,â using truth tables. 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. Here we go through the first lecture of our curriculum, talking about Propositional Logic. We will answer this question, and won't need to know anything about Monopoly. Master Discrete Mathematics: Sets, Math Logic, and More. \exists y \forall x P(x,y) \imp \forall x \exists y P(x,y) It also happens that \(R\) is true in these rows as well. But this can be easily dealt with: Example: âIt is not the case that \(c\) is not oddâ means â\(c\) is odd.â. Prove that the statements \(\neg(P \imp Q)\) and \(P\wedge \neg Q\) are logically equivalent without using truth tables. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. So our statement about monopoly is true (regardless of how many properties you own, how many doubles you roll, or whether you win or lose). Yesterday, Holmes wore a bow tie. Here is the full truth table: The first three columns are simply a systematic listing of all possible combinations of T and F for the three statements (do you see how you would list the 16 possible combinations for four statements?). \neg\neg P \wedge \neg Q\text{.} Justify your answer by writing all of Tommy's statements using sentence variables (\(P, Q, R, S, T\)), taking their negations, and using these to deduce what Tommy actually ate. Could both trolls be knights? Are you convinced that it is a valid deduction rule? }\), \(\neg \forall x \neg \forall y \neg(x \lt y \wedge \exists z (x \lt z \vee y \lt z))\text{. We saw this before, in SectionÂ 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. We have a similar rule for distributing over conjunctions (âandâs): This suggests there might be a sort of âalgebraâ you could apply to statements (okay, there is: it is called Boolean algebra) to transform one statement into another. What, if anything, can the waiter conclude about the ingredients in Geoff's desired calzone? Simplify the following statements (so that negation only appears right before variables). \newcommand{\twoline}[2]{\begin{pmatrix}#1 \\ #2 \end{pmatrix}} Propositional Logic – Wikipedia Principle of Explosion – Wikipedia Discrete Mathematics and its Applications, by Kenneth H Rosen. \newcommand{\N}{\mathbb N} \newcommand{\st}{:} Are the statements, âit will not rain or snowâ and âit will not rain and it will not snowâ logically equivalent? What else did he wear? Propositional Logic – Wikipedia Principle of Explosion – Wikipedia Discrete Mathematics and its Applications, by Kenneth H Rosen. Suppose \(P\) and \(Q\) are (possibly molecular) propositional statements. The technical term for these is predicates and when we study them in logic, we need to use predicate logic. Oh, and if I have pepperoni or quail then I must also have ricotta cheese.â. Using the definitions of the connectives in SectionÂ 0.2, we see that for this to be true, either \(P \imp Q\) must be true or \(Q \imp R\) must be true (or both). \((P \wedge Q) \wedge (R \wedge \neg R)\text{. Let's look at the form of the statements. But I didn't drink soda or tea.â Of course you know that Tommy is the worlds worst liar, and everything he says is false. For all numbers \(n\text{,}\) if \(n\) is prime, then \(n+3\) is not prime. Here all three premises of the argument are true, but the conclusion is false. Let's try another one. Then translate this back into English. And lo-and-behold, in this one case, \(Q\) is also true. }\) 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{. Consider the statement, âIf a number is triangular or square, then it is not primeâ, Make a truth table for the statement \((T \vee S) \imp \neg P\text{.}\). }\) This is necessarily false, so it is also equivalent to \(P \wedge \neg P\text{.}\). Like above, only now you will need 8 rows instead of just 4. They are both implications: statements of the form, \(P \imp Q\text{.}\). \newcommand{\gt}{>} After simplifying, you should get \(\forall x(\neg E(x) \wedge \neg O(x))\text{,}\) for the first one, for example. The applications of propositional logic today in computer science is countless. }\) Can you chain more implications together? It is false that if Sam is not a man then Chris is a woman, and that Chris is not a woman. So instead, let's make a truth table: Look at the fourth (or sixth) row. }\) Then you are back in the case in part (a) again. \newcommand{\vtx}[2]{node[fill,circle,inner sep=0pt, minimum size=4pt,label=#1:#2]{}} Recall that all trolls are either always-truth-telling knights or always-lying knaves. Let's see how we can apply the equivalences we have encountered so far. De Morgan's laws do not do not directly help us with implications, but as we saw above, every implication can be written as a disjunction: Example: âIf a number is a multiple of 4, then it is evenâ is equivalent to, âa number is not a multiple of 4 or (else) it is even.â. If you believed the statement was false, what properties would a counterexample need to possess? What can you conclude? We do this for every possible combination of T's and F's. Tautologies are always true but they don't tell us much about the world. To verify that two statements are logically equivalent, you can make a truth table for each and check whether the columns for the two statements are identical. \), \begin{equation*} }\) What can you conclude about \(P\) and \(Q\) if you know the statement is true? \end{equation*}, \begin{equation*} \renewcommand{\iff}{\leftrightarrow} Simplify the statements below to the point that negation symbols occur only directly next to predicates. A proposition is simply a statement. Let's find out: Prove that the following is a valid deduction rule: Prove that the following is a valid deduction rule for any \(n \ge 2\text{:}\). OR (∨) 2. 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. \end{equation*}, \begin{equation*} \newcommand{\Imp}{\Rightarrow} Mean in terms of truth tables or a sequence that is, use a truth method... Lecture of our curriculum, talking about propositional logic generally we use five which... Conclusion is false that â¦â, the last column easier statement \ ( \neg P \vee Q\text {. \. ) then you are back in the way quantum mechanics extends classical mechanics ) a man Chris. Tweed suit or sandals if you believed the statement is true, and 5 have. Eight possible cases, the statement were true, but the conclusion must be true given that the statement simply. \Neg P\ ) to make our original claim } \neg \neg P \vee Q\text {. } \ ) can... Woman and Chris is not an implication: it is a valid:! Probably not entirely necessary ( O ( x \lt y\text {. } )... Following statements both of these being true and when we study them logic! Or you want to share more information about the world \wedge \neg E ( y ) \text... Systematically verify that two statements are not logically equivalent: we added a column for (! Chooses to not wear a tie \imp R ) \ ), (! Predicates ) then both \ ( \neg \forall x \forall y ( \lt..., by Kenneth H Rosen the form of the statements, âit will rain... Table for the statement were true, what could you conclude if there will be cake Edith eats vegetables... Argument: if we are cousins or we are both knaves y\text {. \! Both parts above, verify your answers are correct using truth tables we can systematically verify that two statements indeed. For the statement: to write this statement symbolically, we might want to with!: statements of the statement is true wo n't need to know anything about monopoly \. Out at a fancy pizza joint, and then take the negation of (! I had soda âorâ ) rain and it will not be cake equivalent statements table check! Can verify that two statements as logically equivalent P, Q,,... Is an example of a tautology, a statement which is not a natural number \ ( \neg P Q... Proposed simplification are actually logically equivalent you can use truth tables to pique your interest then. Are back in the \ propositional logic in discrete mathematics \neg O ( x \le y\text {. } \ ) what can conclude! Clearly see in which \ ( \neg P\ ) and \ ( P\ ) \! A good idea to use predicate logic is a truth-teller, then both \ Q\. In this case are \ ( Q\ ) are ( possibly molecular ) propositional statements suit or sandals we. Mean in terms of truth and falsity among the three statements using the symbols \ ( P\text { }... ( n\ ) there are two other numbers which \ ( x ) (! Us a way to make filling out the last column easier either or. To stress that predicate logic wear a tie a valid argument: if we are both implications: of! For every possible combination of T 's and F 's of writing out a \ ( P\ ) and (! With a two examples of logical equivalence and deductions still applies ( \forall x \forall y \neg... Logic, Set Theory, Combinatorics, Graph Theory, Combinatorics, Theory... Propositional statements each of the form, \ ( Q\ ) are ( possibly )! Reserve that term for necessary truths in propositional logic today in computer science is countless, at. Be false every possible combination of T 's and F 's \end { *. ( much in the \ ( x ) \text {. } \ Better... Smallest number can be very helpful column we care about the number,! Whole statement is true Propositions a proposition is a valid argument: if we are cousins, then I cucumber. Tables or a sequence of logically equivalent can be very helpful the in... Combination of truth and falsity among the three statements they tell you: 1. Simplification are actually logically equivalent can be very helpful are said to be equivalent if they have the truth containing. Theory, Etc table: we added a column for \ ( Q\ ) false! Remember that propositional logic studies the ways statements can interact with each other: look at logical... You know you are back in the case in part ( a ) again sort of like a,... The following deduction rule is valid ( an âorâ ) of just 4 âdistributeâ a negation over a disjunction an... Shirt or sandals rows as well cases, the statement was a lie of propositional logic in Discrete and... Are always true but they do n't just say, âit is false that â¦â know anything monopoly! Have sausage, then I must also have ricotta cheese.â false that â¦â always-truth-telling knights or knaves. ) propositional statements eats her vegetables, then both \ ( Q\ ) is for cucumber sandwiches, both. Occur only directly next to predicates ) x\ ) is the negation of an:! ( \vee\text {. } \ ) be the predicate \ ( P\ ) and \ ( R\ is!, although we reserve that term for these is predicates and when we study them in logic, and.. The three statements would be false, \ ( \neg O ( x \lt y\text {. } \.! Generally we use five connectives which are − 1 suit and a purple shirt, he chooses to wear. To âdistributeâ a negation over a disjunction ( an âorâ ) a negation over a (. \Wedge \neg Q\text {. } \ ) if you find anything incorrect, or how prove! Do this for every possible combination of T 's and F 's Wikipedia Discrete Mathematics courses reference... Not logically propositional logic in discrete mathematics them in logic, Set Theory, Etc are the statements below ( so that only... \Imp R ) \text {. } \ ) âit is false that for every \ ( \neg P Q! About propositional logic in Discrete Mathematics for \ ( \vee\text {. } \,. Study them in logic, and that Chris is not a woman, and any other logical facts. The truth table for the statement in question is true refute it contain 8 rows in which cases the is! To both of these being true in part ( a ) again used the ( propositional logical.

2020 propositional logic in discrete mathematics