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Math 307 Homework: Proofs and Functions - Prof. Daniel Dugger, Assignments of Mathematics

A math homework assignment for a university-level course in discrete mathematics or logic. It includes various problems on logical proofs, set operations, and functions. Students are asked to prove statements using logical rules, determine the image of sets under functions, and apply mathematical induction.

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Pre 2010

Uploaded on 09/17/2009

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Math 307
Homework Due Wednesday, May 20
1. Given RS,P, and PQ, prove that [[Q R]S].
2. Consider the function f:Z7Z7given by f(x) = x3+ 1. Answer the following questions:
(a) Is fone-to-one? Explain why or why not.
(b) Determine f(S) where S={0,2,4,6}.
(c) If A={1,2,3,4}and B={0,4,5,6}, determine If(A) and If(B). Also determine If(AB).
3. In each part, prove the indicated statement by induction:
(a) (nN)[ (2n)!
n!·2nis an odd number]
(b) (1 + 1
2)n>1 + n
2for all n2.
(c) 13+ 23+· ·· +n3= [n(n+1)
2]2for all n1.
4. Let an= 1 2+34 + · · · + (1)n+1n. Prove by induction that a2n=nfor all n1.
5. Consider the sequence given by an=2 + an1and a0= 2. Prove by induction that an2 for all
n0.
6. Consider the sequence given by an= 2an1+ 4an2and initial conditions a0= 0, a1= 3. Prove that
3|anfor all n0.
7. Suppose f:ZZis a function with the property that (x, y Z)[f(x+y) = f(x) + f(y)].
(a) Prove by induction that (nN)[n1(xZ)[f(nx) = n·f(x)]].
(b) Give a line proof that (kN)[f(Mk)Mk].
8. If f:STand g:TU, then there is a function denoted (gf): SUcalled the composition of
gand f. It is defined by the formula
(gf)(x) = g(f(x)).
You can read about compositions on pages 85–87 of your book.
(a) If f:ZZis given by f(x) = x21 and g:ZZis given by g(x) = 3x+ 2, determine (gf)(0)
and (gf)(2). Determine an algebraic formula for (gf)(x) for any integer x.
(b) Suppose f:STand g:TU, and XU. Give a line proof that Igf(U) = Ig(If(U)).
(c) Again suppose that f:STand g:TU. If AS, give a line proof that (gf)(A) = g(f(A)).

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Math 307 Homework Due Wednesday, May 20

  1. Given R ∨ S, ∼ P , and P ⇔ Q, prove that [[Q∨ ∼ R] ⇒ S].
  2. Consider the function f : Z 7 → Z 7 given by f (x) = x^3 + 1. Answer the following questions: (a) Is f one-to-one? Explain why or why not. (b) Determine f (S) where S = { 0 , 2 , 4 , 6 }. (c) If A = { 1 , 2 , 3 , 4 } and B = { 0 , 4 , 5 , 6 }, determine If (A) and If (B). Also determine If (A ∩ B).
  3. In each part, prove the indicated statement by induction: (a) (∀n ∈ N)[ (^) n(2!·n 2 )!n is an odd number] (b) (1 + 12 )n^ > 1 + n 2 for all n ≥ 2. (c) 1^3 + 2^3 + · · · + n^3 = [ n(n 2 +1) ]^2 for all n ≥ 1.
  4. Let an = 1 − 2 + 3 − 4 + · · · + (−1)n+1n. Prove by induction that a 2 n = −n for all n ≥ 1.
  5. Consider the sequence given by an = √2 + an− 1 and a 0 = 2. Prove by induction that an ≤ 2 for all n ≥ 0.
  6. Consider the sequence given by an = 2an− 1 + 4an− 2 and initial conditions a 0 = 0, a 1 = 3. Prove that 3 |an for all n ≥ 0.
  7. Suppose f : Z → Z is a function with the property that (∀x, y ∈ Z)[f (x + y) = f (x) + f (y)]. (a) Prove by induction that (∀n ∈ N)[n ≥ 1 ⇒ (∀x ∈ Z)[f (nx) = n · f (x)]]. (b) Give a line proof that (∀k ∈ N)[f (Mk) ⊆ Mk].
  8. If f : S → T and g : T → U , then there is a function denoted (g ◦ f ) : S → U called the composition of g and f. It is defined by the formula (g ◦ f )(x) = g(f (x)). You can read about compositions on pages 85–87 of your book. (a) If f : Z → Z is given by f (x) = x^2 − 1 and g : Z → Z is given by g(x) = 3x + 2, determine (g ◦ f )(0) and (g ◦ f )(2). Determine an algebraic formula for (g ◦ f )(x) for any integer x. (b) Suppose f : S → T and g : T → U , and X ⊆ U. Give a line proof that Ig◦f (U ) = Ig (If (U )). (c) Again suppose that f : S → T and g : T → U. If A ⊆ S, give a line proof that (g ◦ f )(A) = g(f (A)).