Exam 2

Question 1

A sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables

Answer 1

Predicate

Question 2

The set of all values that may be substituted in place of the variable

Answer 2

Domain

Question 3

Ex:

{x £ D | P(x)}

Answer 3

Truth set

Question 4

Symbol inside down A denotes “for all” and is called:

Answer 4

Universal quantifier

Question 5

A statement of the form “upsidedown Ax £ D, Q(x)”. Defined true if, and only if, Q(x) is true for every x in D.

Answer 5

Universal statement

Question 6

A value for x for which Q(x) is false is called a _____________ to the universal statement

Answer 6

Counterexample

Question 7

The technique used to show the truth of the universal statement:

1^2 >= 1, 2^2 >= 2, 3^2 >= 3, 4^2 >= 4

Hence “upside down x £ D, x^2 >= x” is true.

This is called:

Answer 7

method of exhaustion

Question 8

Consists of showing the truth of the predicate separately for each individual element of the domain.

Answer 8

Method of exhaustion

Question 9

The symbol E denotes “there exists” and is called the:

Answer 9

Existential quantifier

Question 10

Is a statement of the form “Ex £ D such that Q(x).” It is defined as true if, and only if, Q(x) is true for at least one x in D.

Answer 10

Existential statement

Question 11

A reasonable argument can be made that the most important form of statement in mathematics is:

Answer 11

Universal conditional statement

Question 12

The negation of a universal statement (“all are”) is logically equivalent to an:

Answer 12

Existential statement

Question 13

Note that’s when we speak of __________________, we mean that the statements always have identical truth values no matter what predicates are substituted for the predicate symbols and no matter what sets are used for the domains of the predicate variables.

Answer 13

Logical equivalence for quantified statements

Question 14

~(upside down Ax, if P(x) then Q(x)) is logically equivalent Ex such that P(x) and ~Q(x)

Answer 14

Negation of a universal conditional statement

Question 15

The ____________ of a real number a is a real number b such that ab=1.

Answer 15

Reciprocal

Question 16

~(upsidedown A x in D, exists y in E such that P(x,y)) is equivalent to (their exists x in D such that inside down Ay in E, ~P(x,y)

Answer 16

Negations of multiply-quantified statements

Question 17

An integer n is ______ if, and only if, n equals twice some integer.

Symbolically:
Their exists and integer k such that n = 2k

Answer 17

Even

Question 18

An integer n is ______ if, and only if, n equals twice some integer plus 1.

Symbolically:
Their exists and integer k such that n = 2k + 1

Answer 18

Odd

Question 19

An integer n is ______ if, and only if, n > 1 and for all positive integers r and s, if n = rs, then either r or a equals n.

Symbolically:
For all positive integers r and a, is n = rs then either r = 1 and s = n or r= n and s= 1

Answer 19

Prime

Question 20

An integer n is ______ if, and only if, n > 1 and for some integers r and s, with 1

Answer 20

Composite

Question 21

To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property

Answer 21

Method of generalizing from generic particular

Question 22

If the existence of a certain kind of object is assumed or has been deduced then it can be given a name,as long as that name is not currently being used to denote something else

Answer 22

Existential instantiation

Question 23

Star the steps to write a proof:

Answer 23

1. Copy statement of the theorem to be proved on your paper
2. Clearly mark the beginning with “proof”
3. Make you’re proof self-contained
4. write you’re proof in complete, grammatically correct sentences
5. Keep reader informed about status of each statement
6. Give reason for each assertion in proof
7. Display equations and inequalities
8. Include “little words and phrases” that make logic of your arguments clear

Question 24

Common proof mistakes:

Answer 24

1. Arguing from examples
2. using the same letter to mean two different things
3. Jumping to conclusions
4. Circular reasoning
5. Confusion between what is known and what is still to be shown
6. Use of “any” rather than “some”
7. Misuse of the word if

Question 25

A real number r is __________ if and only if, it can be expressed as a quotient of two integers with a nonzero denominator.

Answer 25

Rational

Question 26

The word rational contains the word ratio, which is another word for ____________. A rational number can be written as a ratio of integers

Answer 26

Quotient

Question 27

If neither of the two real numbers is zero, then their product is also not zero

Answer 27

Zero product property

Question 28

It follows that ________ is a quotient if two integers with a nonzero denominator and hence is a rational number

Answer 28

(m+n)/mn

Question 29

The notion of do is ability is the central concept of one of the most beautiful subjects in advanced mathematics: ____________, the study of properties of integers

Answer 29

Number theory

Question 30

If n and d are integers and d != 0 then

Answer 30

n is divisible by d if, and only if n equals d times some integer

Question 31

Instead of “n is divisible by d” we can say that:

Answer 31

n is a multiple of d
d is a factor of n
d is a divisor of n
d divided n

Question 32

One of the most useful properties of divisibility is that it is __________. If one number divides a second and the second divides a third, then the first number divides the third.

Answer 32

Transitive

Question 33

Given any real number x, the _________ of x, denoted [x], is defined as follows:

[x] = that unique integer n such that n <=x

Answer 33

Floor

Question 34

Given any real number x, the _________ of x, denoted [x], is defined as follows:

[x] = that unique integer n such that n -1 < x <=n

Answer 34

Ceiling

Question 35

1. Suppose the statement to be proved is false. That is, suppose that the negation of the statement is true.
2. Show that this supposition leads logically to a contradiction
3. Conclude that the statement to be proved is true

Answer 35

Method of proof by contradiction

Question 36

[PBAC] meaning

Answer 36

Particular but arbitrary chosen

Question 37

What is the definition along rational numbers?

Answer 37

x = a/b and y = c/d and b != 0 and d != 0