Download HW-07 due 03/10

CmSc180 DM Discrete Mathematics
Homework 07 due 03/10
1. Prove by mathematical induction that for all positive integers n
2 + 4 + 6 + .. + 2n = n(n+1)
2. Prove using direct proof that for all integers n, n2 + n is even
Hint: consider two cases: n being even and n being odd
3. Let the universal set be the set R of all real numbers and let
A = {x  R | -5  x < 3}, B = { x  R | 1 < x  6}.
Find each of the following:
AB
=
AB
=
~A
=
~B
=
~A  ~B
=
~A  ~B
=
~(A  B)
=
~(A  B)
=
4. Find the power set of A = { , {a}}
5. Find the first four terms of each of the following sequences defined by an explicit
formula:
a. an = n/(n+2)2, n = 1, 2, 3, ….
b. an = (-1)n+1 2n/(n+1), n = 1, 2, 3, . ..
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c. an = (-2)n-1 /(n+1), n = 1, 2, 3, . ..
6. Find the explicit formula for each of the following sequences:
a. 1 + 1/2, 1/2 + 1/3, 1/3 + 1/4, 1/4 + 1/5, ….
b. 1/4, -2/9, 3/16, -4/25, 5/36, -6/49, …
c. 1/3, 2/4, 3/5, 4/6, 5/7, 6/8, ….
Note: indicate the starting value of n for each formula
7. Determine the properties of the following relations(underline the correct answer)
7.1. A = {1,2,3}, R = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)}
Reflexive
Irreflexive
Neither
Symmetric
Antisymmetric
Neither
Transitive
Not transitive
Is the relation R:
Partial order:
Equivalence relation
yes
yes
no
no
7.2. A = {1,2,3,4}, R = {(1,2), (2,3), (3,4), (4,1)}
Reflexive
Irreflexive
Neither
Symmetric
Antisymmetric
Neither
Transitive
Not transitive
Is the relation R:
Partial order:
Equivalence relation
yes
no
yes
no
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7.3. A = {1,2,3}, R = {(1,1), (1,2), (2,1), (2,2), (3,3)}
Reflexive
Irreflexive
Neither
Symmetric
Antisymmetric
Neither
Transitive
Not transitive
Is the relation R:
Partial order:
Equivalence relation
yes
no
yes
no
8. Consider the relation R defined on the set of all words in the dictionary:
R = {(a,b) | a and b have same number of letters}
Show that this is a relation of equivalence
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