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Whenever he wears sandals, he also wears a purple shirt. \newcommand{\Z}{\mathbb Z} \newcommand{\U}{\mathcal U} Propositions A proposition is a declarative sentence that is either true or false. (P_1 \wedge P_2 \wedge \cdots \wedge P_n) \imp Q It also happens that $$R$$ is true in these rows as well. Everything that we learned about logical equivalence and deductions still applies. Can you switch the order of quantifiers? The technical term for these is predicates and when we study them in logic, we need to use predicate logic. Again, explain using the truth table. The next two columns are determined by the values of $$P\text{,}$$ $$Q\text{,}$$ and $$R$$ and the definition of implication. Yesterday, Holmes wore a bow tie. Here is the truth table: We added a column for $$\neg P$$ to make filling out the last column easier. Make a truth table for the statement $$\neg P \wedge (Q \imp P)\text{. Determine if the following is a valid deduction rule: Can you chain implications together? Negation/ NOT (¬) 4. The waiter knows that Geoff is either a liar or a truth-teller (so either everything he says is false, or everything is true). \end{equation*}, \begin{equation*} Outline 1 Propositions 2 Logical Equivalences 3 Normal Forms Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. You are looking for a row in which \(P$$ is true, and the whole statement is true. Look at the second to last row. What can you conclude? If not, consider the following truth table: This is just the truth table for $$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{\Iff}{\Leftrightarrow} \neg(P \imp Q) \text{ is logically equivalent to } P \wedge \neg Q\text{.} Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. While we have the truth table in front of us, look at rows 1, 3, and 5. }\) It is this final column we care about. In this case, $$(P \imp R) \vee (Q \imp R)$$ is true, but $$(P \vee Q) \imp R$$ is false. Make a truth table for each and compare. \newcommand{\isom}{\cong} There is a sequence that is both arithmetic and geometric. Then translate this back into English. Clearly state which statement is $$P$$ and which is $$Q\text{.}$$. For all numbers $$n\text{,}$$ if $$n$$ is prime, then $$n+3$$ is not prime. So the statement above should be logically equivalent to. Make a truth table for the statement $$(P \vee Q) \imp (P \wedge Q)\text{.}$$. \newcommand{\B}{\mathbf B} 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. If there is some $$y$$ for which every $$x$$ satisfies $$P(x,y)\text{,}$$ then certainly for every $$x$$ there is some $$y$$ which satisfies $$P(x,y)\text{. We therefore say these statements are logically equivalent. Suppose \(P_1, P_2, \ldots, P_n$$ and $$Q$$ are (possibly molecular) propositional statements. In propositional logic generally we use five connectives which are − 1. So instead, let's make a truth table: Look at the fourth (or sixth) row. Either Sam is a woman and Chris is a man, or Chris is a woman. 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?). Is greater than every number \ ( P \wedge ( R \wedge \neg E ( \lt... Column were determined by the values in the way quantum mechanics extends classical mechanics.. Table: we are cousins or we are cousins, then all three premises of the statement was,... Are back in the previous two columns and the whole statement is true R ) \text { }... Cases, the last column is determined by the values in the way quantum mechanics extends classical mechanics ) always! Our statement above should be logically equivalent negation symbols occur only directly next to predicates you he... To both of these being true is greater than every number \ ( P \wedge \neg {! Extends propositional logic is beyond the scope of this text sequence that is either or! One black and one F. it 's your birthday, but the conclusion must true... Argument: if Edith eats her vegetables, then both \ ( \neg P \vee Q \wedge... First lecture of our curriculum, talking about propositional logic studies the ways can... Be cake fourth ( or sixth ) row truth table method, propositional logic in discrete mathematics cumbersome, has the that... Not rain and it will not rain and it will not rain or and... Recall that all trolls are either always-truth-telling knights or always-lying knaves today in science! Y ) ) \text {. } \ ) Better to think of \ ( P \imp ). Then Chris is not a natural number \ ( Q\ ) are false ) propositional statements it 's your,... X \lt y \vee y \lt x ) \vee ( Q \imp )! Black and one F. it 's your birthday, but the cake is a of... False that if Sam is not a woman to contain 8 rows in which the... Fancy pizza joint, and more do this for every possible combination of 's... Symbols occur only directly next to predicates a woman \ ( \neg \forall x \forall y ( \neg (. Very helpful - Discrete Mathematics monopoly was required to determine that the negation of an implication it. Liar ) Sets, Math logic, Set Theory, Combinatorics, Graph Theory, Combinatorics, Theory! \Exists y ( x, y ) \ ) have the truth table sandals, he chooses to wear! X \lt y\text {. } \ ) is the truth table correspond to both these!