homework review notes complete worksheet #1. homework let a = {a,b,c,d}, b = {a,b,c,d,e}, c = {a,d},...
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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
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