# lect4 - DeMorgans Laws (A B) = A B (A B) = A B A B = (A B )...

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1 DeMorgan’s Laws ¬ ( A B ) = ¬ A ∩¬ B ¬ ( A B ) = ¬ A ∪¬ B A B ¬ ( A B ) B ¬ B A ¬ A =

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2 Example of using DeMorgan’s Law Let A = { a, d , e , g } and B = { c , d , f , g } from the universe U = { a , b , c , d , e , f , g , h , i }. To verify that ¬ ( A B ) = ¬ A ∩¬ B find each of these two sets independently to find that they are indeed the same. 1) ¬ ( A B )= ¬ { a, c , d , e , f , g } = { b , h , i } 2) ¬ A ∩¬ B= { b , c , f , h , i } { a , b , e , h , i } ={ b , h , i } are the same, as predicted.
3 Domination Laws A U = U A = Absorption Laws A ( A B ) = A A ( A B ) = A p T T p F F p ( p q ) p p ( p q ) p These laws can be proved by using logic laws or membership tables.

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4 In everyday life different kinds of proofs are acceptable: Jury trial. Word of God. Word of Boss. Experimental science: The truth is guesses and confirmed or refuted by experiments. Sampling: like public opinion is obtained by polling. Inner conviction. . These are not valid proofs in mathematical sense. They all can go wrong… What is a proof? A proof is a method of ascertaining truth.
5 Mathematics uses a particularly convincing way to argue that something is true. Definition . A proof is a formal verification of a proposition by a chain of logical deductions starting from the base set of axioms . A proof takes axioms and definitions and uses deduction rules, step by step, to get a desired conclusion.

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Proof methods If a statement considers a few numbers of cases it can be proved by exhaustive checking . Example : All students in this class are computer science major. We can easily verify is it true or false. Truth table method . To prove a statement about small number Boolean variables make a truth table and check all possible cases. Example : ( p q ) ( ¬ q ¬ p ). By inspection we see that lhs has the same truth value as the rhs
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lect4 - DeMorgans Laws (A B) = A B (A B) = A B A B = (A B )...

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