# Final Exam ```Math 260
Final Exam
Name___________________
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. (20 pts)
Statement
True or
False
Possible Counterexample or Correction
1. P &lt;=&gt; Q means “P iff Q.”
2. Let S = {1,2,3,4,5,6,7,8,9,10}. Then, the
power set Р(S) has 1000 elements.
3. Uc = { }, where U is the Universal set.
4. (PQ) &lt;=&gt; (~P ~Q)
5. h(x)= 1/sin(x) is a well-defined function on
domain .
6.   A, where A is any set and Φ={ }.
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. When defined on domain , f(x) = 10x is
onto  .
9. When defined on , the function
g(x)= x3-x is 1-1 on its range.
10. {x|x=4m, m  }={y| y=8n, n  }
II. Prove or Disprove: P↔Q  (P Q) /\ (PQ) . (10 pts)
III. Perform the following operations. Show work to get credit! (10 pts)
A. Change 78910 to a base 5 number.
B. 110000112 - 111112
Math 260
Final Exam
Name___________________
IV. State and demonstrate three forms of the Fundamental Theorem of Algebra, using the polynomial f(x) =
4x2-4x-3. (12 pts)
V. For the following, suppose U = {1,2,3,4,5,6,7,8,9,10}, S = {3,4}, T = {2,3,5,8,9}, and Φ = { }. Write the
resulting sets using list notation. (Note: U is the universal set of discourse.) (20 pts)
1. (U  S)  TC =
2. S X T =
3. UC =
4. (S  T)c =
5. (U \ T ) \ S =
6. List below the Power Set of T \ S.
7. List below a set of ordered pairs, representing a 1-1 function f with domain S and codomain T.
8. List below a set of ordered pairs representing a function g that is NOT onto, with domain T and codomain S.
9. List below a set of ordered pairs, representing an ONTO function h with domain T and codomain S.
10. List the LARGEST equivalence relation ~ on S.
Math 260
VI. For the following, let x, y 
Final Exam
Name___________________
. Define x ~y &lt;=&gt; xmod4 = ymod4. (15 pts)
A. Prove that ~ is an equivalence relation on
.
B. Describe the equivalence classes for ~ using proper set notation.
C. Prove or disprove: If xmod4 = 3 and ymod4 = 2, then (xy)mod4 = 2.
VII. Prove that .787878… is rational by finding its rational equivalent. You must show your work! (3 pts)
Math 260
Final Exam
Name___________________
VIII. Prove or disprove the following. (10 pts)
A. Let S = {x x  , 6  x} and T = {y y  , 3  y}. Prove or disprove that S  T.
B. Let S = {x x  , 6  x} and T = {y y  , 3  y}. Prove or disprove that T  S.
C. Prove that if x2 + y2 is even , then x + y is even.
Math 260
Final Exam
Name___________________
```