MAT 2070 - Fall 2013 - CityTech
Prof Reitz
Exam 1 Review
Sec 1.1-1.8, 2.1-2.6
** OPTIONAL PROBLEMS: This review sheet is rather long - because of this, certain problems (marked with **) are optional and should be completed only for extra practice.
- Write each of the sets by listing their elements between curly brackets.
b. ** 
c. 
- Write each of the sets in set-builder notation.
- **
- Find the cardinality of each set.
b. 
- Label each statement True or False, and include a brief explanation.
a. 
b. **
c. 
For the remaining parts, let 
and 
d. 
e. 
f. **
- Given sets
and universal set 
, find each of the following and state the cardinality:
a. 
b. 
c. 
d. 
e.** 
f.** 
g. 
h. 
i.** 
j.** 
k.
- Given intervals
, and 
,
Write in interval notation: a.** 
b. 
c. 
Sketch in the plane: d. 
e.** 
- Venn diagrams.
- Sketch a Venn diagram for:
- Sketch a Venn diagram for:
Write an expression for each of the Venn diagrams below.
c.** d.
- Find the union and intersection of each collection.
- Let
, 
, 
, and in general for each 
, 
. Find 
and 
. - For each , let
be the closed interval of real numbers 
. Find and . - For each real number
, let 
. Find 
and 
(give a description in words and a sketch). - ** For each real number
, let 
. Find 
and 
(give a description in words and a sketch).
- For each item, determine if it is a statement. If so, determine whether it is True or False. If not, is it an open sentence?
- Frogs have yellow blood.
.- Add 7 to both sides of the equation.
- Is
?
- Write a truth table for each statement (be sure to include intermediate steps as necessary).
- **
- Translate the given statement S into logical form, using statements P, Q and R. Then write a truth table for the result.
S: If 
and x is not a perfect square then 
.
P:
Q: x is a perfect square
R:
- Determine whether each pair of statements is logically equivalent by comparing rows in a truth table.
and 
- **
and 
- Express (in English) the contrapositive of each statement.
- If a frog can jump, then it’s alive.
- If x is a two-digit even number, then x isn’t prime.
Exam 1 Review ANSWER KEY
If you discover an error please let me know, either in class, on the OpenLab, or by email to jreitz@citytech.cuny.edu. Corrections will be posted on the “Exam Reviews” page.
- a. { -1, 2, 7, 14, 23, … } b. { -4, -3, -2, -1, 0, 1, 2, 3, 4 } c. { -22, -15, -8, -1, 6, 13, 20 }
- a.
b. 
c. 
d. 
- a. 4 b. 5
- a. T - the only element of the set on the left,
, is also an element of the set on the right.
b. F - the set “
” does not appear inside the set on the right.
c. T - the set is a subset of
, so it is an element of 
d. T - is a subset of D, so it is an element of 
e. F -
is an element of 
, NOT of 
f. F - We check to see if the set on the left is a subset of. For this to be true, each member should be an ordered pair with the first element from D and second element from E. 
fits this description, but 
does not. - a.
, the cardinality is 4 b. 
, 1
c.
, 8 d. 
, 3 e. 
, 7
f.
, 2 g. 
, 6 h. 
, 2
i.
, 4
j.
, 16
k.
, 2 - a.
b. 
c. 
d. e. - a. b.
c. There are many possible solutions, including
d. There are many possible solutions,
including
, and 
- a.
and 
b.
and 
c.
and 
d.
and 
- a. A statement, False b. Not a statement, an open sentence
c. Not a statement, not an open sentence
d. A statement, False e. Not a statement, not an open sentence - a.
P | Q |
|
|
|
T | T | F | T | T |
T | F | F | T | T |
F | T | T | F | T |
F | F | F | T | T |
b.
P | Q | R |
|
|
| |
|
T | T | T | T | T | T | T | T |
T | T | F | T | F | F | F | F |
T | F | T | F | T | F | T | F |
T | F | F | F | F | F | F | T |
F | T | T | F | F | T | T | F |
F | T | F | F | F | F | F | T |
F | F | T | F | F | F | F | T |
F | F | F | F | F | F | F | T |
- a.
b.
P | Q | R |
|
|
T | T | T | F | T |
T | T | F | F | T |
T | F | T | T | T |
T | F | F | F | T |
F | T | T | F | T |
F | T | F | F | T |
F | F | T | T | F |
F | F | F | F | T |
- a. Yes, they are logically equivalent.
P | Q |
|
|
T | T | T | T |
T | F | F | F |
F | T | T | T |
F | F | T | T |
b. Yes, they are logically equivalent
P | Q | R |
|
|
|
|
|
T | T | T | T | T | T | T | T |
T | T | F | T | T | T | T | T |
T | F | T | F | F | T | F | F |
T | F | F | F | T | T | T | T |
F | T | T | F | F | F | T | F |
F | T | F | F | T | T | T | T |
F | F | T | F | F | F | F | F |
F | F | F | F | T | T | T | T |
- a. If a frog is not alive, then it can’t jump.
b. If x is prime, then it is not a two-digit even number.