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:Z7→Z7given 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(A∩B).
3. In each part, prove the indicated statement by induction:
(a) (∀n∈N)[ (2n)!
n!·2nis an odd number]
(b) (1 + 1
2)n>1 + n
2for all n≥2.
(c) 13+ 23+· ·· +n3= [n(n+1)
2]2for all n≥1.
4. Let an= 1 −2+3−4 + · · · + (−1)n+1n. Prove by induction that a2n=−nfor all n≥1.
5. Consider the sequence given by an=√2 + an−1and a0= 2. Prove by induction that an≤2 for all
n≥0.
6. Consider the sequence given by an= 2an−1+ 4an−2and initial conditions a0= 0, a1= 3. Prove that
3|anfor all n≥0.
7. Suppose f:Z→Zis 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→Tand g:T→U, then there is a function denoted (g◦f): S→Ucalled the composition of
gand 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→Zis given by f(x) = x2−1 and g:Z→Zis 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→Tand g:T→U, and X⊆U. Give a line proof that Ig◦f(U) = Ig(If(U)).
(c) Again suppose that f:S→Tand g:T→U. If A⊆S, give a line proof that (g◦f)(A) = g(f(A)).