Download 1. Is there a real number whose square is a

Math 263 Discrete Structures
1.
Exam #1
Name:_______________________________________
Is there a real number whose square is −1?
a. Is there a real number x such that ____?
b. Does there exist ____ such that x2 = −1?
2. Given any two real numbers, there is a real number in
between.
a. Given any two real numbers a and b, there is a real number
c such that c is ____.
b. For any two ____, ____ such that a < c < b.
3. Rewrite the following statements less formally, without using variables. Determine whether the statements are
true or false.
a. There are real numbers u and v with the property that u +v < u − v.
b. For all positive integers n, n2 ≥ n.
4. Which of the following sets are equal?
A = {a, b, c, d} B = {d, e, a, c}
C = {d, b, a, c} D = {a, a, d, e, c, e}
5.
a. Is 4 = {4}?
b. How many elements are in the set {3, 4, 3, 5}?
c. How many elements are in the set {1, {1}, {1, {1}}}?
6. Use the set-roster notation to indicate the elements in each
of the following sets.
a.
S = {n ∈ Z| n = (−1)k , for some integer k}.
b. U = {r ∈ Z| 2 ≤ r ≤ −2}
b.
V = {s ∈ Z| s > 2 or s < 3}
7. Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation
R from A to B as follows: For all (x, y) ∈ A × B, (x, y) ∈ R means that
y/x is an integer.
a.
Is 4 R 6? Is 4 R 8? Is (3, 8) ∈ R? Is (2, 10) ∈ R?
b.
Write R as a set of ordered pairs.
c.
Write the domain and co-domain of R.
d.
Draw an arrow diagram for R.
8. Represent the common form of each argument using letters to stand for component sentences, and fill in the
blanks so that the argument in part (b) has the same logical form as the argument in part (a).
a. If all integers are rational, then the number 1 is rational.
All integers are rational.
Therefore, the number 1 is rational.
c.
If all algebraic expressions can be written in prefix notation, then
____________________________________________________________ .
.
Therefore, (a + 2b)(a2 − b) can be written in prefix notation.
9. Rewrite the following statement in if-then form: Freeze or I’ll shoot.
____________________________________________________________
10. Use modus ponens or modus tollens to fill in the blanks in the argument so as to produce valid inferences.
If √2 is rational, then √2 = a/b for some integers a and b.
It is not true that √2 = a/b for some integers a and b.
∴ .________________________________________________