KA
CHAPTER 2 TEST
* Fill in the blanks using a variable or variables to rewrite the given statement.
1. Is there a real number whose square root is – 1?
a. Is there a real number x such that ? b. Does there exist — such that Vx = -1?
8. Let B = {2, 4, 6, 8, 10}, C = {4, 8, 10}, and D = {.1
1 x is even}. Answer the following questions. Give reasons for your answers.
a. Is D s B? b. Is C s D? c. Is C S B?
d. Is B a proper subset of D? 9. a. Is ((−1), 12) = (1?, (–192)? Explain.
2. Given any real number, there is a real number that
is lesser. a. Given any real number r, there is — s such
that s is - b. For any — , - such that s b. 18 (176.4)=(* 1.)? Explain.
c. Is (–22,0)=(-V16,0)? Explain.
Fill in the blanks to rewrite the given statement. 3. For all real numbers x, if x is an integer then x is a
rational number. a. If a real number is an integer, then —. b. For all integers x, c. If x — , then — . d. All integers x are
10. Let A = {1, 2, 3, 4} and B = {0, 1}. Use the set
roster notation to write each of the following sets, and indicate the number of elements that are in each set: а. Ах В b. BxA c. AXA d. BxB
4. All real numbers have squares that are not equal to
-1. a. Every real number has — b. For all real numbers r, there is for r c. For all real numbers r, there is a real number s
such that
11. Let C = {0, 1, 2} and D = {2, 4, 6, 8) and define
a relation R from A to B as follows: For all (x, y) E AXB,
(x, y) e R means that Y72 is an integer.
5. There is a positive integer whose square is equal to
itself. a. Some — has the property that its — . b. There is a real number r such that the square of
ris c. There is a real number r with the property that
for every real number s -
a. Is 1 R 2? Is 2 R8? Is (1,8) € R? Is (2,6) ER? 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.
6. a. Let A be the set containing all prime numbers
less than 30. List down the elements of A. b. Is {2, 2} = {2, {2}}? c. How many elements are in the set {a, a, a, a, a)?
12. Define a relation A from R to R as follows: For all
(x, y) e RxR, (x, y) € A means that x y a. Is 57 A 53? Is (-17) A (-14)? Is (14, 14) € A?
Is (-35, 1) E A? b. Draw the graph of A in the Cartesian plane.
7. Given that Z denotes the set of all intégers and N.
the set of all natural numbers, describe each of the following sets. a. {xe Nix < 10 and x is divisible by 3} b. {x € Zlx is prime and x is divisible by 2} c. {x S Zlr? = 4}
13. a. Find all relations from {a, b, c} to {u „v}.
b. Find all functions from {a, b, c} to {u ,v}. c. What fraction of the relations from (a, b, c) to
(u ,v} are functions?
14. Let X = {a, b, c) and Y = {1, 2, 3, 4]. Define a
function F from X to Y by the arrow diagram below.
bo
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a. Write the domain and co-domain of F. b. Find F(a), F(b), and F(c). c. Represent F as a set of ordered pairs.
15. Let A = {0, 1, 2, 3} and define functions F and
G from A to A by the following formulas: For all xe A, F(x) = (x+4) and G(x) = (x2 +3x+1). Is F = G? Explain.