74-hw6

# 74-hw6 - g is injective then g ◦ f is injective 2(b Prove...

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MATH 74 HOMEWORK 6 - DUE MONDAY, MARCH 12 1. Fix n N , and let A n = { x N : 1 x n } . The answers to 1(a), 1(b), 1(c) should all be given in terms of n . [You may, if you ﬁnd it necessary, break the problem into various cases, provided that you clearly state what they are, and what the answer is in each case.] 1(a). How many subsets does A n have? [A proof of this was on last week’s homework, so you only need to state the answer.] 1(b). How many subsets of A n have an odd number of elements? How about an even number of elements? [Don’t just state the answer; prove it.] 1(c). How many subsets of A n contain at least one odd number? [Again, prove your answer.] Extra credit. How many subsets of A 20 do not contain a consecutive pair of numbers? Can you ﬁnd a formula valid for general n ? 2-4. Let A, B, C be sets, and f : A B and g : B C be functions. The composite function g f is the function from A C whose rule is given by x 7→ g ( f ( x )). 2(a). Prove, using only the deﬁnition of “injective,” that if f is injective and
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Unformatted text preview: g is injective then g ◦ f is injective. 2(b). Prove, using only the deﬁnition of “surjective,” that if f is surjective and g is surjective then g ◦ f is surjective. 3(a). If g ◦ f is surjective, is g necessarily surjective? Prove or disprove. 3(b). If g ◦ f is surjective, is f necessarily surjective? Prove or disprove. 4(a). If g ◦ f is injective, is g necessarily injective? Prove or disprove. 4(b). If g ◦ f is injective, is f necessarily injective? Prove or disprove. 5. What sort of things would you like the most preparation for, in the remaining time we have available? • Algebra of various sorts [prep for 110/113/114] • Number theory [prep for 113/115] • Calculus/analysis [prep for 104/105/185] • Geometry [prep for 130/140/142/. ..] If you have no interest in a subject, include that also. Feel free to add in any other comments on the class. 1...
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