Homework Review notes Complete Worksheet #2. Homework State whether the conditional sentence is true or false 1.If 1 = 0, then 1 = – 1 True F F T

Homework

• Review notes• Complete Worksheet #2

Homework

State whether the conditional sentence is true or false

1. If 1 = 0, then 1 = – 1 True

F F T

Homework

Give the converse of the conditional sentence and state if it is sometimes, always, or never true.

5. If two triangles are congruent, then their corresponding angles are congruent.

If the corresponding angles of two triangles are congruent, then the triangles are congruent. Sometimes true.

Homework

Give the converse of the conditional sentence and state if it is sometimes, always, or never true.

9. If ab < ac, then b < c

If b < c, then ab < ac, sometimes true

Homework

State the contrapositive for each conditional sentence.

13. If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

If a parallelogram is not a rectangle, then no one of its angles is a right angle.

Homework

State the assumptions that would have to be made if the given statement is to be proven by contradiction.

17. 2 2If , then .x y x y 2 2Assume that and .x y x y

Homework

State each conditional sentence in the if-then form.

21. For all and , is necessary for .R S R S R R S

If , then .R S R R S

Homework

State each conditional sentence in the if-then form.

25. For 0, 0, 0, is neccessary for .m ma b m a b a b

If 0, 0, 0, and , then .m ma b m a b a b

Homework

Name the axiom, theorem, or definition that justifies each step.

1.

Proof:

Definition of Subtraction

Associative Axiom

Additive Inverses

Additive Identity

Transitive Property

xyyx

yyxyyx x y y

0x x

xyyx

Homework

Solve over

5. .

7 3 2 2 3 8x x

7 21 4 6 8

17 8

25

x x

x

x

Homework

Solve over

9. .

26 5 1 0x x

3 1 2 1 0

13 1 0 3 1 3

1 1 12 1 0 2 1 ,2 2 3

x x

x x x

x x x x

Homework

State whether each set is closed under (a) addition and (b) multiplication. If not, give an example.

13. N

(a) Closed

(b) Closed

Conditional SentencesAddition and Multiplication Properties of Real Numbers

Foundations of Real Analysis

Order

The real numbers are ordered by the relation less than (<). a < b if the graph of a is to the left of the graph of b on the number line. Less than (<) is an undefined relation.

Axiom of Comparison

For all real numbers a and b, , , or a b a b a b

Transitive Axiom of Order

For all real numbers a, b and c, if a < b and b < c, then a < c

Addition Axiom of Order

For all real numbers a, b and c, if a < b, then a + c < b + c

Positive and Negative

A real number, x, is positive if x > 0 and negative if x < 0

Multiplication Axiom of Order

For all real numbers a, b and c, if 0 < c and a < b, then ac < bc.

Sign Definitions

• a > b if and only if b < a• a ≥ b if and only if a > b or a = b• a ≤ b if and only if a < b or a = b

Theorem Three

For all real numbers a, b, and c:1. If and , then .a b c d a c b d

12. If 0, then 0 and 0.a a

a

13. If 0, then 0 and 0.a a

a

Theorem Four

For all real numbers a, b, and c, if c < 0 and a < b, then ac > bc

Theorem Five

For all real number a and b:• ab > 0 if and only if a and b are both positive or a and b are

both negative.• ab < 0 if and only if a and b are opposite in sign.

Example

Solve the inequality and graph each non-empty solution.

2.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

4422 t

Example


6.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

54620 t

Example


10.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

322 nn

Example


14.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

0144 2 rr

Example

18. Solve and graph the intersection of the sets:

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

4 7 13

3 2 17

x

x

Absolute Value

Definition – For all real numbers x:

This may also be written as a piecewise function:

if 0

if 0

x x x

x x x

, 0

, 0

x xx

x x

For exampe 3 3, and 3 3 3

Theorem Six

If a ≥ 0, |x| = a if and only if x = –a or x = a.

If a > 0, |x| < a if and only if –a < x < a

If a > 0, |x| > a if and only if x < –a or x > a

Theorem Seven

For all real numbers a, |a|2 = a2

Theorem Eight

For all real numbers a and b,1. ab a b

2. aa

b b

3. a b a b

4. a b a b

Example

Find and graph each non-empty set over R.

2.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

523 y

Example


6.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

22

1t

Example


10.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

079 c

Example


18.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

rr 3238

Example


22.

-----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----

1 21

2 y

Example

Solve.

26. 2 1 3x

Natural Numbers

The set of all positive integers {1, 2, 3,…}

Factor or Divisor

a is a factor of b (symbolically, a|b) if there is an integer c such that ac = b

Prime Number

Any number greater than 1 that has only one and itself as factors. Sometimes called a prime.

Composite

Any integer greater than 1 that is not prime

Theorem Nine

The Fundamental Theorem of Arithmetic – Every integer greater than 1 can be expressed as a product p1p2p3…pn in which p1, p2, p3, …, pn are primes. Furthermore, the factorizing is unique, except for the order in which primes are written.

Greatest Common Factor

The greatest common factor or GCF of a and b is the largest positive integer that is a factor of both a and b

Relatively Prime

Integers whose GCF is 1

Theorem Ten

The Division Algorithm – Given integers s and t, t > 0, there exist unique integers q and r such that s = tq + r and 0 ≤ r < t

Euclidean Algorithm

To find the GCF, divide the larger by the smaller, then the divisor by the remainder and so on until the remainder is zero; the last divisor is the GCF

For example:198

3 3654

541 18

36

362 0

18

Rational numbers

All numbers that may be expressed in the form , 0a

bb

Irrational Numbers

All real numbers that are not rational, e.g., π and e, the base for natural logarithms

Theorem Eleven

There is no rational number whose square is 2.

Least Common Multiple

The least common multiple or LCM is the smallest possible integer that is a multiple of both integers

Perfect Number

A natural number equal to the sum of all of its factors other than itself

Example

Find the prime factors of each number.

2. 1960

Example

Find the prime factors of each number.

6. 508

Example

Find the GCF of each pair of integers

10. 216; 539

Example

Find the LCM for each pair.

14. 1000; 120

Example

Use the Euclidean algorithm to find the GCF.

18. 33, 9

Example

State whether the integer is perfect.

22. 126

Homework

• Complete Worksheet #3• Test on Real Analysis next time

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Homework Review notes Complete Worksheet #2. Homework State whether the conditional sentence is true or false 1.If 1 = 0, then 1 = – 1 True F F T