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Homework 02

# Homework 02 - MATH 74 HOMEWORK 2 1 Due Wednesday February...

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MATH 74 HOMEWORK 2 1 Due Wednesday February 4th at 3:10pm. (1) (a) Define the statement ‘ f : R R is continuous at a R .’ (b) Write the negation of the statement in (a), without using ‘not.’ (c) Choose an example of a function that is not continuous at a point and prove that it is discon- tinuous using the definition from (b). (2) (Bergman 6) Suppose that A and B are sets. Translate each of the statements below, which are expressed using the symbols and into a statement about A and B expressed using only the set-theoretic symbols discussed in class ( , , , , , c , etc.). (a) ( x )(( x A ) = ( x B )) (b) ( x )(( x A ) ⇐⇒ ( x B )) (c) ( x )( x / A ) (d) ( x )(( x / A ) ( x B )). (3) (Bergman 9) Consider the sentence, “There is someone at the hotel who cleans every room.” Explain two ways this sentence can be interpreted and translate them into two quantifications of the relation “X cleans Y.” (4) Let S = [0 , 2] and let T = { 6 /n | n Z and n 1 } . Compute, with proof, ( S × T ) ( Z × Z ). (5) Prove the 2nd De Morgan Law: ( A B ) c = A c B c . (6) Determine (with proof) if the following functions are injective, surjective, both or neither.
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