homework1mat25 - A is a collection of sets B satisfying(a...

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Homework 1 due Thursday August 7th 1. Construct a truth table for the statement: [ p ( q ∧ ∼ q )] ⇔ ∼ p. 2. Consider the following statement: A function f is uniformly continuous on a set S if and only if for every > 0 there is a δ > 0 such that | f ( x ) - f ( y ) | < whenever x and y are in S and | x - y | < δ . (a) Rewrite the statement using the logical symbols , , , , etc. (b) Write the negation of the statement in words and in logical symbols. 3. Prove or give a counterexample: For every positive integer n , n 2 + 3 n + 8 is even. 4. Prove: A B if and only if A B = A . 5. An open cover
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Unformatted text preview: A is a collection of sets B satisfying: (a) Each B ∈ B is open. (b) A is contained in the union of sets in B , i.e. A ⊆ [ B ∈B B . Show that B = ±² 1 n , n-1 n ³ for n = 0 , 1 , 2 ,... ´ is an open cover of (0 , 1). Is there a finite subset of B that is an open cover of (0 , 1)? 6. Prove that when A and B are countable, A × B is countable. 7. Let f : A → B and C,D ⊆ B . Prove f-1 ( C ∪ D ) = f-1 ( C ) ∪ f-1 ( D ). What about f-1 ( C ∩ D )? 1...
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