Math 260
Test II
Name___________________(1pt)
I. Directions: For each statement write “True” or “False” in the column to the right. If an answer is false,
provide a counterexample or correction, where applicable. (30 pts)
Statement
True or
False
Possible Counterexample or Correction
1. With domain , f(x) = x3-x is 1-1 with its
codomain .
2. With domain , f(x) = x3 is 1-1 with
codomain .
3. f(x) = x + sin x has real-valued function
with domain .
4. With domain , f(x) = 2x+3 is onto its
codomain .
2
5. With domain *, f(x) = is onto its
x
codomain .
6. Suppose set D is both domain and
codomain for the relation r = {(x,x): x  D };
then r is a function.
7. Suppose set D is both domain and
codomain for the relation r = {(x,x): x  D };
then r is an equivalence relation on D.
8. Suppose f(x) = x3-x is a function with
domain . Define x~y if and only if
f(x) = f (y). Then 0,1, and 2 are in the same
equivalence class.
9. r = xmod10 means that 10 = mx +r for
some integer m.
10. x is an even integer if and only if x2 is
even.
II. Write the following statements in symbolic form; then PROVE or DISPROVE (counterexample) NEATLY! (45 pts)
A. Let a,b, c be integers. If a does NOT divide b and if a does NOT divide (b+c), then a does NOT divide c.
Symbolic Form:
Proof or Counterexample:
Math 260
Test II
Name___________________(1pt)
B. Let x be any integer. If xmod10 = 6, then 10 divides (x2-x).
Symbolic Form:
Proof or Counterexample:
C. All polynomials p(x) = ao+a1x1+a2x2+…aixi+…+anxn with real coefficients ai have real roots.
Symbolic Form: You do NOT need this part.
Proof or Counterexample:
III. State and demonstrate three forms of the Fundamental Theorem of Algebra, using the polynomial f(x) =
6x2-x-1. (24 pts)
Math 260
Test II
Name___________________(1pt)