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{\Huge 189-235A: Basic Algebra I} \\ \sk
{\Huge Assignment 1} \\ \sk
{\Large Due: Monday, September 24 }
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1. Use Cardano's formula to solve the following cubic equations. In each case
say how many real solutions there are and list all such solutions when there are
more than one. (You are advised to use a calculator to check that the
expressions
you've written down are indeed solutions to the equation at hand.)
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a. $x^3+3x+1$
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b. $x^3-3x+1$. (In this case, give a closed form expression for the solution(s)
of the equation, in terms of $\cos(2\pi/9)$ and $\sin(2\pi/9)$.)
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2. Let $S$ and $T$ be the sets $\{ a,b, c\}$and
$\{x,y\}$ respectively.
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a. How many functions are there from $S$ to $T$?
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b. How many injective
functions?
\noindent b.
How many surjective functions?
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3. Let $X$ be a set, and let ${\cal F}(X)$ be the set of all functions from
$X$ to itself. This set is equipped with a natural binary operation
$(f,g) \mapsto fg$ , given
by the composition of functions.
\noindent a. Show that $f(gh) = (fg)h $ for all $f$, $g$, $h$ in ${\cal F}(X)$.
(In other words, the operation of composition of funcions is {\em associative}.)
\noindent b. Show, by providing an example,
that $fg$ need not be equal to $gf$, i.e., that composition of functions
{\em need not be commutative}.
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4. Show ({\em without resorting to a calculator or computer}!)
that the complex number
$(1+\sqrt{3}i)^{111} $ is an integer, and write it down in factored form.
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5. Show using induction that for all $n\ge 1$,
$$ 1^3+\cdots + n^3 = (1+\cdots +n)^2.$$
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6. Using the Euclidean algorithm compute the gcd of
$910091$ and $3619$. Show the steps in your calculation.
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7. Using induction (or otherwise) show that $7$ divides $8^n-1$ for
all $n\ge 0$.
Use induction to show that $49$ divides
$8^n-7n-1$ for all $n\ge 0$.
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8. Using induction, show that the addition law in $\mathbf N$ is associative
directly from the axioms defining addition in $\mathbf N$.
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9. Exercise (17), page 32 of the on-line notes.
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10. Exercise (19), page 33 of the on-line notes.
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