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exercises_2 - Analysis 1 Exercises 2 1 Let A B be sets from...

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Analysis 1: Exercises 2 1. Let A, B be sets from a universe. Prove the following properties of the complement. (a) ( A c ) c = A . (b) ( A B ) c = A c B c . (c) ( A B ) c = A c B c . 2. Let A, B and C be sets. Prove the following equalities (a) A 4 B = ( A B ) - ( A B ). (b) ( A 4 B = ) ( A = B ). 3. What do the following statements mean? Do you think they true or false? (a) ( x N )( y N )( x < y ). (b) ( y N )( x N )( x < y ). (c) ( x N )( y N )( x < y ). (d) ( y N )( x N )( x < y ). (e) ( x N )( y N )( x < y ). (f) ( x N )( y N )( x < y ). 4. Analyse the logical form of the following statements. (a) Jane saw a bear and Roger saw one too. (b) Jane saw a bear and Roger saw it too. You could use, for example, P ( x, y ) to stand for the proposition ” x saw y ”. 5. Let P ( x ) be the statement that says that a real number x has some property P . Rewrite the following propositions using connectives and quantifiers. Construct negations to each of them.
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