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CS275 Discrete Mathematics
Gongbo “Tony” LiangFourth year PhD student in [email protected]@cs.uky.edu
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Section 2.4Sequences and Summations
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Sequence
A sequence is a discrete structure used to represent an ordered list.- E.g. ,
- {an} with an=3n
- n=0, an=1
- n=1, an=3
- n=2, an=9
- n=3, an=27
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Find these terms of the sequence {an}, where an = 2•(-3)n+5n
- a0- a1- a4- a5
3
-1
787
2639
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List the first 10 terms of :
- The sequence that begins with 2 and in which
each successive term is 3 more than the preceding term
- {an} with an = 2+3n
- 2,5,8,11,14,17,20,23,26,29
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List the first 4 terms of:
- The sequence whose nth term is n!-2n
(start with n=1)
- {an} with an = n!-2n
- -1, -2, -2, 8
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Find the simple formula or rule that generates the terms of an integer sequence that begins with the given list
- 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, …
- One 1 and one 0, followed by two 1s and 2 0s,
followed by three 1s and three 0s, …
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Find the simple formula or rule that generates the terms of an integer sequence that begins with the given list
- 1,2,2,3,4,4,5,6,6,7,8,8,…
- The positive integers are listed in increasing order
with each even positive integer listed twice
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What is the value of each of these sums of terms of a geometric progression
- ∑#$%& 3 • 2#
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What is the value of each of these sums of terms of a geometric progression
- ∑#$%& 3 • 2#- j = 0, 3 • 2#=3•20 = 3•1 = 3
- j = 1, 3 • 2#=3•21 = 3•2 = 6
- j = 2, 3 • 2#=3•22 = 3•4 = 12
- j = 3, 3 • 2#=3•23 = 3•8 = 24
- j = 4, 3 • 2#=3•24 = 3•16 = 48
- ∑#$%& 3 • 2# = 3+6+12+24+48+96+192+384+768 = 1355
- j = 5 3 • 2#=3•25 = 3•32 = 96
- j = 6, 3 • 2#=3•26 = 3•64 = 192
- j = 7, 3 • 2#=3•27 = 3•128 = 384
- j = 8, 3 • 2#=3•28 = 3•256 = 768
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What is the value of each of these sums of terms of a geometric progression
- ∑#$%& 2 • (−3)#
- 9842
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Section 2.5Cardinality of Sets
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Cardinality of sets
- Cardinality of finite sets:
- The number of elements in the set
- Countable sets:
- A set that either finite or has the same cardinality as the set of positive integers
- If S is a infinite set and S is countable
- |S|=א 0 (aleph null)
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Is “the odd negative integers” countable?If so, exhibit a one-to-one correspondence between the set of positive integers and that this set.
- Yes, it is countable.
- 1<-> -1- 2 <-> -3
- 3 <-> -5- …
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Is “integers not divisible by 3” countable?If so, exhibit a one-to-one correspondence between the set of positive integers and that this set.
- Yes, it is countable.
- 1<-> 1- 2 <-> -1
- 3 <-> 2- 4 <-> -2 ...
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Show that a finite group of guests arriving at
Hilbert’s fully occupied Grand Hotel can be given
rooms without evicting any current guest.
- Hilbert’s Grand Hotel has countable infinite number of
rooms
- Suppose m new guests arrive at the fully occupied hotel. We can move the guest in Room n to Room m+nfor n=1,2,3…; then the new guests can occupy rooms 1 to m.
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Given an example of two uncountable sets A and B such that A∩B is:a) finite, b) countable infinite, c) uncountable
- a) A = [1,2], B = [3,4]
- b) A = [1,2]∪Z+, B = [3,4]∪Z+
- c) A = [1,3], B = [2,4]
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Section 5.1Mathematical Induction
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Mathematical induction can be used to prove statements such as “P (n) is true for all positive integers n”
- General Steps:
- Basis step: We verify P(1) is true
- Inductive step: For all positive integers k, we assume P(k)
is true and show P(k + 1) is true
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Let P(n) be the statement that 12+22+…+n2 = n(n+1)(2n+1)/6 for the positive integer n.
- A) what is the statement P(1)?- B) Show that P(1) is true
- C) What is the inductive hypothesis?- D) Complete the inductive step
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Let P(n) be the statement that 12+22+…+n2 = n(n+1)(2n+1)/6 for the positive integer n.
- A) what is the statement P(1)?- P(1) = 1
- B) Show that P(1) is true- Left side: P(1) = 12=1
- Right side: P(1) = 1(1+1)(2+1)/6=1- Left side = Right side- P(1) is true
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Let P(n) be the statement that 12+22+…+n2 = n(n+1)(2n+1)/6 for the positive integer n.
- C) What is the inductive hypothesis?- 12+22+…+k2 = k(k+1)(2k+1)/6
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Let P(n) be the statement that 12+22+…+n2 = n(n+1)(2n+1)/6 for the positive integer n.
- D) Complete the inductive step
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Prove that if A1, A2, …, An and B are sets, than ("# ∪ "% ∪…∪ "&) ∩ )
= ("# ∩ )) ∪ ("% ∩ )) …∪ ("& ∩ ))- Let P(n) be “ "# ∪ "% ∪ … ∪ "& ∩ )= ("# ∩ )) ∪ ("% ∩ ))…
∪ ("& ∩ )) ”
- Basic step: * # = "# ∩ ) = "& ∩ )- Inductive step:
- …
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Prove that if A1, A2, …, An and B are sets, than ("# ∪ "% ∪…∪ "&) ∩ )
= ("# ∩ )) ∪ ("% ∩ )) …∪ ("& ∩ ))- Inductive step: