MATH15Aquiz4sol

# MATH15Aquiz4sol - ⊆ B ∩ C . ] Suppose that x is an...

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Name (Last, First) SOLUTIONS Please circle your section: A01 A02 A03 A04 5 PM 6 PM 7 PM 8 PM Math 15A Quiz 4 Briggs Fall Quarter 2004 No books, notes, calculators, or any unauthorized assistance is permitted on this quiz. Read each question carefully, answer each question completely, and show all of your work. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If a question is not clear, ask for clariﬁcation. GOOD LUCK! 1. (5.1 #21; 12 points ) A Venn diagram has been supplied for each of parts (a) - (f). For each part, shade the region corresponding to the indicated set. a. A B b. B C U A B C A U C B c. A C d. A - ( B C ) B C U A A B C U e. ( A B ) C f. Note by DeMorgan’s: ( A B ) C = A C B C U B A C U B A C QUIZ CONTINUES ON THE BACK 1

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2. (5.2 #12; 10 points ) Using an element argument, prove that for all sets A , B , and C , if A B , then A C B C . Assume that all sets are subsets of a universal set U . Proof: Suppose that A , B , and C are sets for which A B . [ We would like to show that A C
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Unformatted text preview: ⊆ B ∩ C . ] Suppose that x is an element in A ∩ C . By deﬁnition of intersection of sets, it follows that x ∈ A and x ∈ C . But by the assumption that A ⊆ B , since x ∈ A , x ∈ B . Therefore x ∈ B and x ∈ C which implies by deﬁnition of the intersection of sets that x ∈ B ∩ C . Thus, A ∩ C ⊆ B ∩ C . 3. (7.2 #21; 8 points ) Deﬁne Floor : R → Z by the formula Floor( x ) = b x c , for all real numbers x . (a) Is Floor one-to-one? Prove or give a counterexample. (b) Is Floor onto? Prove or give a counterexample. Solution: (a) Floor is NOT one-to-one since Floor(1) = b 1 c = 1 = b 1 . 5 c = Floor(1 . 5) while 1 6 = 1 . 5. (b) Floor : R → Z is onto since if n ∈ Z , Floor( n ) = b n c = n . That is, since the ﬂoor of any integer is the integer itself, we can say that Floor maps R ONTO the set of integers. 2...
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## This note was uploaded on 04/14/2008 for the course MATH 15A taught by Professor Briggs during the Fall '04 term at UCSD.

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MATH15Aquiz4sol - ⊆ B ∩ C . ] Suppose that x is an...

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