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Logic http://localhost/~senning/courses/ma229/slides/logic/slide01.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide02.html Logic prev | slides | next prev | slides | next Propositions A proposition is a statement that is

  1. Logic http://localhost/~senning/courses/ma229/slides/logic/slide01.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide02.html Logic prev | slides | next prev | slides | next Propositions A proposition is a statement that is either true or false, but not both. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Logic 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:33 PM 1 of 1 08/26/2003 03:33 PM Logic http://localhost/~senning/courses/ma229/slides/logic/slide03.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide04.html Logic Logic prev | slides | next prev | slides | next Propositions: Example Propositions The following statements are propositions: We usually use lowercase letters for propositions: 2 + 2 = 4 p, q, r, ... I am an American my hair is blue Let p be a proposition. The negation of p is written p (or p’ ) and is read "not p ." The following statements are not propositions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 6 + 8 Simon says "sit down" Do you want to go to the store? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:33 PM 1 of 1 08/26/2003 03:34 PM

  2. Logic http://localhost/~senning/courses/ma229/slides/logic/slide05.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide06.html Logic Logic prev | slides | next prev | slides | next Truth Tables Conjunctions A truth table is a useful tool with which to analyse propositions. It Let p and q be propositions. The proposition " p and q ", denoted p works by listing all possible truth values of a set of propositions. q , is true when p and q are both true, otherwise it is false. All possible truth values of the proposition p are listed in the truth We say that p q is the conjunction of p and q . table p : eyes are blue q : hair is brown p p t f p q : eyes are blue and hair is brown. f t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM Logic http://localhost/~senning/courses/ma229/slides/logic/slide07.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide08.html Logic Logic prev | slides | next prev | slides | next Disjunctions Truth Tables for Conjunctions and Disjunctions Let p and q be propositions. The proposition " p or q ", denoted p q , conjunction disjunction is false when p and q are both false and true otherwise. p q p q p q p q We say that p q is the disjunction of p and q . t t t t t t t f f t f t p : eyes are blue q : hair is brown f t f f t t f f f f f f p q : eyes are blue or hair is brown. Notice how the number of rows depends on the number of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 propositions. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM

  3. Logic http://localhost/~senning/courses/ma229/slides/logic/slide09.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide10.html Logic Logic prev | slides | next prev | slides | next Exclusive Or Exclusive Or There are two different types of "or" operations in logic. Let p and q be propositions. The proposition denoted p xor q is the exclusive or of p and q and is true when exactly one of p and q is 1. If you’ve had MA141 or PH121 then you are good at true and the other is false. mathematics. exclusive or 2. You may choose the pizza or the porkchop. p xor q p q The first of these is the disjunction form. The second "or" is an t t f exclusive or ; you can have one or the other but not both. t f t f t t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 f f f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM Logic http://localhost/~senning/courses/ma229/slides/logic/slide11.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide12.html Logic Logic prev | slides | next prev | slides | next Implications Implications Implications are also called conditionals and are usually read "if p Let p and q be propositions. The implication p q is the then q ." proposition that is false when p is true and q is false and is true otherwise. It is read " p implies q ." Consider the following statement: Here p is the hypothesis and q is the conclusion . if I can write then I can read. p q implication p q p q When is this true? When is this false? Is it always either true or false? t t t t f f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 f t t f f t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM

  4. Logic http://localhost/~senning/courses/ma229/slides/logic/slide13.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide14.html Logic Logic prev | slides | next prev | slides | next Biconditionals Converse and Contrapositive Consider the compound proposition The converse of p q is q p . Note that the the truth value of an implication and its converse are not necessarily the same. ( p q ) ( q p ) The contrapositive of p q is q p . Note that these have the This is called a biconditional and is read " p if and only if q ." We same truth values. denote this proposition with " p q ." p q p q p q q p q p p q p q q p p q t t f f t t t t t t t t t f f t f t f t f f t f f t t f t f t f t t f f f f t t t t t f f t t t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM Logic http://localhost/~senning/courses/ma229/slides/logic/slide15.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide16.html Logic Logic prev | slides | next prev | slides | next Propositional Equivalences Propositional Equivalences Key Idea A compound proposition is one involving more than one simple proposition. For example p q is a compound proposition made up Replace one proposition by another that is equivalent in order to of the two simple propositions p and q . achieve a desireable goal. A compound proposition is a tautology if it is always true and it is Example: p can always be replaced by ( p ) since a truth table a contradiction if it is always false; otherwise it is called a indicates that both p and ( p ) have the same truth values. contingency . The propositions p and q are logically equivalent if p q is a p p ( p ) tautology. We use the notation p q to denote logical equivalence. t f t f t f 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM

  5. Logic http://localhost/~senning/courses/ma229/slides/logic/slide17.html Logic http://localhost/~senning/courses/ma229/slides/logic/slide18.html Logic Logic prev | slides | next prev | slides | next Exercises Logical Equivalences Use truth tables to show the following: Equivalence Name p T p 1. p ( q r ) ( p q ) ( p r ). Identity laws p F p 2. p q is logically equivalent to ( p q ) ( p q ). p T T Domination laws 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 p F F p p p Idempotent laws p p p Double negation law ( p) p 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM 1 of 1 08/26/2003 03:34 PM Logic http://localhost/~senning/courses/ma229/slides/logic/slide19.html Logic prev | slides | next Logical Equivalences Equivalence Name p q q p Commutative laws p q q p ( p q ) r p ( q r ) Associative laws ( p q ) r p ( q r ) p ( q r ) ( p q ) ( p r ) Distributive laws p ( q r ) ( p q ) ( p r ) ( p q ) p q De Morgan’s laws ( p q ) p q 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 of 1 08/26/2003 03:34 PM

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