HomeworkReview notesComplete Worksheet #1

HomeworkLet A = {a,b,c,d}, B = {a,b,c,d,e}, C = {a,d}, D = {b, c}

Describe any subset relationships.

1. A; D

D A

HomeworkLet E = {even integers}, O = {odd integers},

Z = {all integers}

Find each union, intersection, or complement.

5. E

E Z E O

HomeworkState whether each statement is true or false.

9. - False A square all parallelograms

HomeworkIf A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:

13.

 

A B

1,2,3,4 1,4,6,8 1,2,3,4,6,8

1,2,3,4,6,8

A B

A B

HomeworkIf A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:

17. A’

 

1,2,3,4,5,6,7,8,9,10,11,12,13 1,2,3,4

5,6,7,8,9,10,11,12,13

5,6,7,8,9,10,11,12,13

A U A

A

HomeworkIf A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:

21.

 

A B C

1,2,3,4 1,4,6,8 1,2,3,4,6,8

1,3,6,7,9,11,12,13

1,2,3,4,6,8 1,3,6,7,9,11,12,13

1,3,6

1,3,6

A B

C

A B C

A B C

HomeworkIf A = {1, 2, 3, 4}, B = {1, 4, 6, 8}, C = {2, 4, 5, 8, 10} and U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, find:

25. A

1,2,3,4 1,2,3,4

1,2,3,4

A

A

HomeworkList all subsets of each set.

29. {4}

, 4

HomeworkThe power set of a set A, denoted by P (A) is the set of all subsets of A. Tell how many members the power set of each set has.

33. {4}

The power set of A has 21 = 2 members

HomeworkState whether each statement is true or false.

1. 4 is an even number and 5 is an odd number – True

 

HomeworkFind and graph each solution set over R; i.e., p, q, and

p Λ q

5. p: x > 0; q: 2x < 6 → p: x > 0; q: x < 3

ο-------------ο

ο---→ ←---ο

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

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

HomeworkFind and graph each solution set over R; i.e., p, q, and p Λ q

9 p: 4t – 5 ≥ 3; q: 3t + 5 ≤ 26 → p: 4t ≥ 8; q: 3t ≤ 21 →

p: t ≥ 2; q: t ≤ 7

 

●---------------------●

●------→ ←------●

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

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

HomeworkFind and graph each solution set over R; i.e., p, q, and

p ν q

13. p: 3w – 1 > 5; q: 4w +3 ≤ -1 → p: 3w > 6; q: 4w ≤ - 4 → p: w > 2; q: w ≤ -1

←------● ο------→

←------● ο------→

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

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

HomeworkWrite the negation of each sentence.

17. There is a positive square root of 2.

There is not a positive square root of 2.

HomeworkWrite the negation of each sentence.

21. 6 6 12

23 3 6

and

6 6 122

3 3 6or

Homework25. Find and graph on a number line the solution set over R of the negation of the conjunction

2x < -4 or 3x > 6 → 2x ≥ -4 and 3x ≤ 6 → x ≥ -2 and x ≤ 2

●------------●

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

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

 

HomeworkState whether each sentence over is an identity, a contradiction, or a sentence that is sometimes true and sometimes false.

29.

 

22 2 1 1x x x

Sometimes true 0 and sometimes false 0 .x x

HomeworkState whether each sentence over is an identity, a contradiction, or a sentence that is sometimes true and sometimes false.

33.

Sometimes true 0 and sometimes false 0 .x x

2x x

Conditional SentencesAddition and Multiplication Properties of Real Numbers

Foundations of Real Analysis

Conditional SentenceConditional sentence – sentence in which there is a dependency of one sentence on another; if p and q are sentences, a conditional sentence relating them is “if p, then q” (p → q)

Conditional sentences, by definition, are always true except when p is true and q is false

Converse – the opposite dependency of a conditional sentence, the converse of p → q is q → p (“if q, then p”)

Biconditional sentences are true only when both p and q are true or both p and q are false

Contrapositive – statement q’ → p’ is the contrapositive of p → q

Example #1State whether the conditional sentence is true or false

2. If 12 is a multiple of 6, then 24 is a multiple of 6

 

Example #2Give the converse of the conditional sentence and state if it is sometimes, always, or never true.

6. If 2 is a factor of an integer, then 2 is a factor of the square of that integer.

 

Example #3Give the converse of the conditional sentence and state if it is sometimes, always, or never true.

10. If x2 < 0, then x4 ≥ 0

 

 

Example #4State the contrapositive for each conditional sentence.

14. If ab = ac and a ≠ 0, then b = c.

Formal Mathematical SystemsA formal mathematical system consists of:

Undefined objects Postulates or axioms Definitions Theorems

Axioms of EqualityAxioms of Equality (for all real numbers a, b, and c) :

Reflexive Property: a = a Symmetric Property: If a = b, then b = a Transitive Property: If a = b and b = c, then a = c

Substitution AxiomSubstitution Axiom: If a = b, then in any true sentence involving a, we may substitute b for a, and obtain another true sentence

Axioms of Addition Closure For all real numbers a and b, a + b is a unique real number

Associative For all real numbers a, b, and c

Additive Identity There exists a unique real number 0 (zero) such that for every real number a.

Additive Inverses For each real number a, there exists a real number – a (the opposite of a) such that

Commutative For all real numbers a and b,

a b c a b c

aaa 00

0 aaaa

abba

Axioms of Multiplication Closure For all real numbers a and b, ab is a unique real number

Associative For all real numbers a, b, and c

Multiplicative Identity There exists a unique real number 1 (one) such that for every real number a.

Multiplicative Inverses For each real number a, there exists a real number (the reciprocal of a) such that

Commutative For all real numbers a and b,  

bcacab

aaa 11

a

1

111

a

aaa

baab

Distributive Axiom of Multiplication over AdditionFor all real numbers a, b, and c, acabcba

DefinitionsSubtraction :

Division: provided b ≠ 0

baba 1a

ab b

Theorem One For all real numbers a, b, and c:

1. a = b if and only if a + c = b + c Cancellation Law of Addition

2. a = b if and only if ac = bc (c ≠ 0) Cancellation Law of Multiplication

3. If a = b, – a = – b

4. – ( – a) = a

5. a∙0 = 0

6. – 0 = 0

7. – a = – 1(a)

Theorem One Continued For all real numbers a, b, and c:

8. – ab = a (– b) = – a (b)

9. – (a + b) = – a + ( – b)

10. If a ≠ 0, aaa

111

Theorem Two For all real numbers a and b: ab = 0 if and only if a = 0 and/or b = 0

Example #5Name the axiom, theorem, or definition that justifies each step.

2. If a = b, then a2 = b2

Proof:

a = b

aa = ab

ab = bb

aa = bb

a2 = b2

Example #6Solve over R.

6. 0413 xx

Example #7Solve over R.

10. 0537 2 xx

Example #7State whether each set is closed under (a) addition and (b) multiplication. If not, give an example.

14. {1}

HomeworkReview notesComplete Worksheet #2