Test 1

Question 1

What are the Fundamental properties of the set of real numbers under + and x

Answer 1

1. The Closure Properties
2. Commentative Properties
3. Associative Properties
   4.Existence of Identities
4. Existence of Inverse
5. Distributive Property
6. Cancellation Properties

Question 2

What is the closure property?

Answer 2

If a and b are real numbers, then a + b and ab are (real
numbers, integers, rational numbers, natural numbers).

Question 3

What is the Commentative Property?

Answer 3

If a and b are real numbers, then a + b = b + a
and ab = ba.

Question 4

What is the Associative Property?

Answer 4

If a, b, and c are real numbers, then (a + b) + c = a + (b + c) and (ab)c = a(bc).

Question 5

What is the Existence of Identities?

Answer 5

(i) 0 is a (real number, integer, natural number, rational number) for which a + 0 = 0 + a = a for all (real numbers, integers, natural numbers, rational number) a.
(ii) 1 is a (real number, integer, natural number, rational number) for which 1 x a = a x 1= a for all (real numbers, integers, natural numbers, rational numbers) a.

Question 6

What is the Existence of Inverses?

Answer 6

(i) If a is any (real number, integer, natural number, rational number), then there exists a (real number, integer, natural number, rational number) -a such that a + (-a) = (-a) + a = 0.
(ii) If a is a nonzero real number, then there exists a real number 1/a such that a x 1/a = 1/a x a = 1.
(ii integers) If a is an integer, then there exists an integer d such that a x d = d x a = 1 if and only if a = ±1.

Question 7

What is the distributive property?

Answer 7

If a, b, and c are real numbers, then a(b + c) = ab + ac.

Question 8

What is the cancellation Properties?

Answer 8

(i) If a, b, and c are real numbers such that a + b = a + c, then b = c.
(ii) If a, b, and c are real numbers such that ab = ac and a ≠ 0 , then b = c.

Question 9

What are some fundamental properties of the set of integers under + and x

Answer 9

1. Closure Property
2. Existence of Identities
3. Existence of Inverses

Question 10

What are some fundamental properties of the set of rational numbers under + and x

Answer 10

1. Closure Property
2. Existence of Identities
3. Existence of Inverses

Question 11

What are the Order Structures on the set of real numbers?

Answer 11

1. Trichotomy Property: If a and b are real numbers, precisely one of the following statements holds: a < b, b < a, or a = b.
2. Antisymmetry Property: If a and b are real numbers such that a ≤ b and b ≥ a , then a = b.
3. (i) If a, b, and c are real numbers such that a < b, then a + c < b + c.
   (ii) If a, b, and c are real numbers such that a ≤ b , then a + c ≤ b+ c .
4. (i) If a, b, and c are real numbers such that a < b and 0 < c, then ac < bc.
   (ii) If a, b, and c are real numbers such that a ≤ b and 0 ≤ c , then ac ≤ bc .
5. (i) If a, b, and c are real numbers such that a < b and c < 0, then bc < ac.
   (ii) If a, b, and c are real numbers such that a ≤ b and c ≤ 0 , then bc ≤ ac .
6. Transitive Properties:
   (i) If a, b, and c are real numbers such that a < b and
   b < c, then a < c.
   (ii) If a, b, and c are real numbers such that a ≤ b and
   b ≤ c , then a ≤ c .

Question 12

What is the theorem for Division Algorithm for Integers?

Answer 12

If a and b are integers with
b ≠ 0 , then there exist integers q and r such that a = bq + r and 0 ! r < | b |.

Question 13

What is the definition of a rational number?

Answer 13

a real number a is a rational number if there exist
integers c and d with a = c
d and d ≠ 0 .

Question 14

What is an integer?

Answer 14

a number that is not a fraction; a whole number.

Question 15

What is a natural number?

Answer 15

a positive whole number (1, 2, 3, etc.), sometimes with the inclusion of zero.

Question 16

What is the definition of a even and odd number?

Answer 16

m is said to be even if there exists an integer n such that m = 2n. On the other hand, m is said to be odd if there exists an integer n such that m = 2n + 1.

Question 17

What letters are used for real numbers, rational numbers, integers, natural numbers?

Answer 17

R to denote the set of real numbers, Q to denote the set of rational numbers, Z to denote the set {0,±1,±2,±3,…} of integers, and N to denote the set {1,2,3,…} of natural numbers.

Question 18

What is the definition of divisibility?

Answer 18

If m and n are integers with n ≠ 0, we say that n divides m (and m is a multiple of n) if there exists an integer c such that m = cn. In this case we say that n is a divisor of m.

Question 19

What is the definition of a prime number?

Answer 19

An integer p is said to be prime if p > 1 and the only positive divisors of p are 1 and p itself.

Question 20

What is the truth table for P and Q?

Answer 20

P Q P ∧ Q ~Q (P∧Q) ∨ (~Q)
T T T F T
T F F T T
F T F F F
F F F T T

Question 21

What are the properties of propositions?

Answer 21

a) P v (~P) is a tautology.
b) De Morgan’s Laws:
(i) ~(P v Q) is equivalent to (~P)∧ (~Q).
(ii) ~(P∧ Q) is equivalent to (~P)v (~Q).
c) ~(~P) is equivalent to P.
d) Associative Properties:
(i) (P∧ Q)∧ R is equivalent to P∧ (Q∧ R).
(ii) (P v Q)v R is equivalent to Pv (Q v R).

Question 22

What is a denial?

Answer 22

Let P be a proposition. A denial of P is any proposition equivalent to ~P. So, for
example, by Theorem 1.1(b), (~P) ∧ (~Q) is a denial of P v Q since (~P)∧ (~Q) is
equivalent to ~(P v Q).

Question 23

What is a contrapositive?

Answer 23

3. The contrapositive of P→Q is the conditional statement (~Q)→(~P)

Question 24

What is the Universal Quantifier?

Answer 24

(∀x)P(x) , read “for all x, P(x),” is true when P(x) is true for all values of x in the universe for the open sentence P(x) .

Question 25

What is the Existential Quantifier?

Answer 25

(∃x)P(x) , read “there exists an x such that P(x) ,” is
true if the truth set for P(x) is nonempty.