Section 1.3
SEQUENCES
A sequence is a list of objects in a definite order.
The sequence may be finite:
0, 1, 2, 3, 2, 1, 0
The sequence may be infinite:
2, 4, 6, 8, 10,
3, 9, 27, 81, 243,
SEQUENCES CONTINUED
Recursive sequence refers to the previous ter
1.2
UNION
Union is a set that consists of all elements that belong to A or B.
Let set A = cfw_ 1, 2, 3, 5, 7
Let set B = cfw_ 2, 4, 6, 8, 9
2
A B = cfw_ 1, 2, 3, 4, 5, 6, 7, 8, 9
Notice that we dont repeat any elements
INTERSECTION
Intersection is a set
Sets and Subsets
SETS
Any well defined collection of objects is
called a set. The members of the collection
are called elements of the set.
Examples of sets:
Collection of people that like to play soccer.
Collection of dogs with white fur.
Collection of b
Jennifer Klarfeld, Section 4.3-4.4, Edition 8
SECTION 4.3
5. Prove that if : is onto and : is onto, then : is onto.
By Theorem 4.2.1, = = .
For all , because is onto, there exists such that = .
Since
Jennifer Klarfeld Sec 5.2, 5.3 Edition 8
SECTION 5.2
3. Prove that the following sets are denumerable.
(b) , the positive integer multiples of 3.
To show that is denumerable, we must show that .
Le
Jennifer Klarfeld Homework 9 Section 3.3/3.5 8th Edition
Section 3.3
2. For the given set , determine whether is a partition of .
(a) = cfw_, , , , = cfw_, , cfw_, , cfw_,
i. For all sets in , the set is nonempty.
ii.
Let = cfw_, and = cfw_, . Since an
Jennifer Klarfeld Sec. 5.1 Edition 8
1. Prove Theorem 5.1.1: Show that the relation is reflexive, symmetric, and transitive
on the class of all sets.
Let , and
Jennifer Klarfeld Homework 8 Sec 3.2 Edition 8
1
1. Indicate which of the following relations on the given sets are reflexive on
the given set, which are symmetric, and which are transitive.
(c) on N
(i)
(ii)
(iii)
For all x N , x is equal itself . So=isr
Jennifer Klarfeld Homework Section 1.6-2.1
Section 1.6
1. Prove that:
(b) there exist integers and such that + = .
= 1 = 1.
15(1) + 12(1) = 15 12 = 3.
,
15 + 12 = 3, = 1 = 1.
(d) there do not exist integers and s
Jennifer Klarfeld Homework 9 Section 3.3/3.5 8th Edition
Section 3.3
2. For the given set A , determine whether P is a partition of A .
(a) A=cfw_1,2, 3, 4 , P=cfw_1,2,cfw_2,3 , cfw_3,4
i. For all sets in P , the set is nonempty.
ii.
Let X =cfw_1,2 and
Jennifer Klarfeld Homework Section 1.6-2.1
Section 1.6
1. Prove that:
(b) there exist integers m and n such that 15 m+12 n=3 .
Let m=1n=1.
Then15 (1)+12(1)=1512=3.
Therefore ,there exist integers mn such that
15 m+12 n=3, namely m=1n=1.
(d) there do not e
Jennifer Klarfeld Sec. 5.1 Edition 8
1. Prove Theorem 5.1.1: Show that the relation is reflexive, symmetric, and
transitive on the class of all sets.
Let A , B and C be sets in the class of all sets, and let be the relation
equivalence of sets.
(i)
For al
Jennifer Klarfeld Homework #5 Sec. 2.2
7. For all sets A, B, and C:
(a) .
Let .
Then it is true that either or .
Thus, .
Since implies , then by definition of a subset, .
(e) = .
Jennifer Klarfeld
Let X be a set.
Then XX, since every set is a subset of itself.
Therefore, R is reflexive.
Let X and Y be sets, and suppose XY.
It is not necessarily so that XY implies YX.
Consider the counterexample X=cfw_1, 2 and Y=cfw_1, 2, 3.
XY,
Jennifer Klarfeld Homework #3 Section1.4-1.5
Section 1.4
6. Let a and be be real numbers. Prove that:
(a) |ab|=|a|b|.
Let a and b be real numbers.
Case 1: Let a 0 and b 0. Then ab 0, and |ab| = ab = |a|b|.
Case 2: Let a 0 and b 0. Then ab 0, and |ab| = ab
Jennifer Klarfeld Week 10 Edition 8
6.
Section 4.1
(b) Let A be the set cfw_1,2,3, and let R be the relation on A given by cfw_ (x,y) : 3x+y is
prime. Prove that R is a function with domain A.
If x = 1, then 3x+y = 3+y. For y = 1, 2, or 3, 3+y is prime if
Jennifer Klarfeld Homework 7 Ed. 8 Sec. 2.5, 2.2, 3.1
Section 2.5
5.
(a) Show that at n months, there are pairs of rabbits.
At 1 month, there is 1 pair of rabbits, which is equal to so the statement is true for n=1.
At 2 months, there is still
Jennifer Klarfeld
T
T
F
F
T
F
T
F
~
F
T
F
T
~
T
T
F
T
Consider = 0.
Because zero does not have a multiplicative inverse, then there does not exist a value such
that 0 = 1.
Therefore, the stat
Jennifer Klarfeld Midterm 03/06/2016
1. Explain whether ~( ) and ~ ~ are equivalent.
T
T
F
F
T
F
T
F
T
F
F
F
~( )
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F
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~ ~
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F
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For
Jennifer Klarfeld, Homework 6, Sec 2.3 and Sec 2.4
Section 2.3
1(j). Find the union of the intersection of:
= , = , , , ,
3
=
4
3
= cfw_
4
1
Jennifer Klarfeld, Hom
Jennifer Klarfeld - Homework #1 Section 1.1 Edition 7
3. Make truth tables for each of the following propositional forms.
(a)
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F
(h) Q
Q
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(k) PP
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1
Jennifer Klarfeld - H
Jennifer Klarfeld
Let be a real number such that " > 16.
Then " > 16, which means that 4 < < 4.
Therefore, = 5 is a counterexample to the above statement.
We will prove this statement by contradiction.
Let be rational and be ir
Jennifer Klarfeld Week 10 Edition 8
6.
Section 4.1
(b) Let A be the set cfw_1,2,3, and let R be the relation on A given by cfw_ (x,y) : 3x+y is
prime. Prove that R is a function with domain A.
If x =
Jennifer Klarfeld Midterm 03/06/2016
1. Explain whether
(P Q) and
P
Q
P Q
T
T
F
F
T
F
T
F
T
F
F
F
P Q
are equivalent.
( P Q)
F
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T
P
Q
F
F
T
T
F
T
F
T
P Q
T
T
F
T
For these two statements to be equivalent, their truth tables must be the
same. In other w
Jennifer Klarfeld Homework 8 Sec 3.2 Edition 8
1
1. Indicate which of the following relations on the given sets are reflexive on
the given set, which are symmetric, and which are transitive.
(c) =
(i
Jennifer Klarfeld Final Exam
1. For propositions , , and , prove that ~( ) and ~ are equivalent.
~( )
~
~
T
T
T
F
F
F
F
T
F
T
T
T
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F
F
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As we
Jennifer Klarfeld Homework #3 Section1.4-1.5
Section 1.4
6. Let a and be be real numbers. Prove that:
(a) |ab|=|a|b|.
Let a and b be real numbers.
Case 1: Let a 0 and b 0. Then ab 0, and |ab|
Jennifer Klarfeld Homework 7 Ed. 8 Sec. 2.5, 2.2, 3.1
Section 2.5
5.
(a) Show that at n months, there are f n pairs of rabbits.
At 1 month, there is 1 pair of rabbits, which is equal to
f1
so the statement is true for
n=1.
At 2 months, there is still 1 pa