Math161 HW#1, DUE Date: 13/11/09, 14.00 hours 1) For U = {1, 2,3, 4,5,6,7,8,9,10} let A = {1, 2,3,4,5} , B = {1,2, 4,8}, C = {1, 2,3,5,7}, and D = {2, 4,6,8} . Determine each of the following: (a) (A∪B)∩C={1,2,3,5} (b) A∪(B∩C)={1,2,3,4,5}
(c) C ∪ D ={1,3,4,5,6,7,8,9,10} (d) (A∪B)−C={4,8} (e) A∪(B−C)={1,2,3,4,5,8}
(f) (B −C)−D={} (g) B−(C−D)={2,4,8}
(h) (A∪B)−(C ∩D)={1,3,4,5,8}
(i) A ⊕ B ={3,5,8}
2) Given that U=
A=
B=
{x | x is a positive integer less than 20}
{1,5,9,19}
{b | b is a positive odd integer less than 11}={1,3,5,7,9}
{c | c is a positive odd integer less than 20}={1,3,5,7,9,11,13,15,17,19}
C=
a) A∪B ={1,3,5,7,9,19}
b) A∩B∩C={1,5,9}
c) (A∪C)′={2,4,6,8,10,12,14,16,18}
d) (A∪B)∩C={1,3,5,7,9,19}
3) The universal set is U={1,2,3,4,5,6,7,8,9,10,11}.Let A={1, 2,3, 4,5,6}, B = {2, 4,6,8,10},
C ={3, 4,5,7,9,11}. Find the followings.
a) A∪B={} b) B −(A∪C)={8,10}
c)(B − A)∪C={3,4,5,7,9,10,11} d) A∩C={2,4,6} e) A′={7,8,9,10,11}
f) Draw the VENN DIAGRAM of these sets and find (A∪B)−C and B′ . 4) Given the Universal set U={positive integers not larger than 12}, and the sets A={positive integers not more than 6} B={3,4,6,7} C={5,6,7,8,9,10} Find ii) (A U B) /C ={1,2,3,4}
i) AU B ={1,2,3,4,5,6,7}
iii) | A−B |=3
iv) P(A‐B)=Power set of (A‐B)={ф,1,2,5,{1,2},{1,5},{2,5},{1,2,5}}
5) Let A = {1, 2,3, 4,5} and B = {1,3,5}.
a) List all subsets of A that are disjoint from B .
{ ф ,2,4,{2,4}}
b) List at least three subsets of A that are incomparable with B .
use the definition of incomparable sets…
c) How many proper subsets of B are there?
n(B)=2, 2n-1=7
d) Find n(A∪B)=5
6) In a survey of 100 students, the following data was collected. There were 19 taking
biology, 20 taking chemistry, 19 taking physics, 7 taking physics and chemistry, 8 taking
biology and chemistry, 9 taking biology and physics, and 5 taking all three subjects.
a) How many of the group were not taking any of three subjects? 61
b) How many were taking only chemistry? 10
c) How many were taking physics and chemistry, but not biology? 20
7) How many bit strings with length 8 either start with 1 or end with 00?
|A1| = number of length 8 bitstrings that start with 1 = 27
|A2| = number of length 8 bitstrings that end with 00 = 26
5
|A2 ∪ A1| = number of length 8 bitstrings that end with 00 and start with 1 = 2
7
6
5
Thus: |A1 ∪ A2| = 2 + 2 – 2
9) Let X = {1,2,3,4,5,6,7,8,9,10,12,14,16}and let R be a relation on X defined as
R = {(a,b) : a − b is a multiple of 5}.R={(1,6),(6,1),(2,7, (7,2), (3,8), (8,3) ,(4,9), (9,4), (5,10),
(10,5), (7,12), (12,7), (9,14), (14,9)}
(a) Show that R is an equivalence relation.
(i) (a, a ) ∈ R, ∀a ∈ X since a-a=o is multiple of 5 , R is reflexive
(ii) if (a, b ) ∈ R, then ( b, a ) ∈ R, a-b=5k then b-a=-(a-b)= - 5k is multiple of 5 , R is symmetry
(iii) if
( a, b ) ∈ R, and ( b, c ) ∈ R, then ( a, c ) ∈ R, a-b=5k1 and b-c=5k2 then (a-b)+(b-c)=(a-c)=
5k1 + 5k2 =5(k1 + k2) is multiple of 5 , R is transitive
R is equivalence relation
(b) Write the distinct equivalence classes.
C(0), C(1), C(2), C(3), C(4)
10) Let Z be the set of integers.For any two numbers a,b in Z, if a – b is even(divisible by 2).
a) Show that this is an equivalence relation
b) How many equivalence classes does this equivalence relation have? list them all.
C(0)={...-4,-2,0,2,4..}
C(1)={...,-3,-1,1,3,...}
11) Let Z be the set of integers. For any two numbers a,b in Z. if a – b is divisible by n.
a) Show that this is an equivalence relation
b) How many equivalence classes does this equivalence relation have?
There are n equivalence classes
12) Define ~ on Z by a ~b iff 3a+b is multiple of 4
a) Prove that ~ defines an equivalence relation
b) Find the equivalence class of 0
c) Find the equivalence class of 2
13) For any a and b, define a~b if 3a+4b=7n for some integer n
a) Prove that ~ defines an equivalence relation
b) Find the equivalence class of 0
14) Given A = {1,2,3,4}. Consider the relation on A ,R = {(1,1),(2,2),(3,3),(4,4),(1,2),(1,3)(2,3),(3,2),(2,1),(3,1)}
a) Determine whether the R is reflexive? Explain.
R is reflexive, because (1,1),(2, 2),(3,3),(4, 4)∈R .
b) Determine whether the R is symmetric? Explain.
R is symmetric, because (1,2),(2,1)∈R, (2,3),(3,2)∈R
c) Determine whether the R is transitive? Explain.
R is transitive, because (1, 2),(2,1)∈R, (1,1)∈R (1,3),(3,2)∈R, (1,2)∈R
d) Determine whether the R is antisymmetric? Explain.
R is not antisymmetric, because (1,2)∈R, (2,1)∈R
f)
15) Let X={1,2,3}and Y={a,b,c,d}. Define H : X →Y as follows: H(1)=c, H(2)=a and H(3)=d .
16) If A ={a,b,c} , B ={x, y}and C = {u,v,w}and if f : A→ B and g : B→C are the
functions f ={(a, x),(b, y),(c, x)} and g ={(x,u),( y,w)}.Find the composition go f .
go f={(a,u),(b,w),(c,u)}
17) Let A={1,2,3,4} , B={a,b,c,d,e} and C={7,8,9} . Consider f : A→ B and g : B→C as
defined by
f ={(1,a),(2,b),(3,d),(4,b)} and g ={(a,8),(b,9),(c,8)(d,7),(e,9)}.
Find g o f and f o g, if defined, if not defined explain why.
18) If f and g are the functions f: R→ R defined by f(x)=2x − 3 and g(x)=x2 +1
Find g o f and f o g
g o f= (2x-3)2 +1 = 4x2-12x+9+1 = 4x2-12x+10
f o g=2(x2 +1)-3 = 2x2 +2-3 = 2x2 -1
1
19)Given f ( x) = 2 x − 3 and g ( x) = ( x + 3)
2
a) Find f o g ( 3) , g o f ( 2 )
f ( g (3)) = f (3) = 3
g ( f (3)) = g (3) = 3
b) Are they inverses of each other? if they are find I(5).
⎛1
⎞
fog ( x) = 2⎜ ( x + 3)⎟ − 3 = x + 3 − 3 = x
⎝2
⎠
1
gof ( x) = (2 x − 3 + 3) = x
yes they are inverses
2
I (5) = fog (5) = 5
20) Let A={ x ∈ R | x ≠ 2 } and B={ x ∈ R | x ≠ 1 }. Define f : A → B and f : B → A by
x
2x
f ( x) =
g ( x) =
x−2
x −1
Q) Show that the function f : R → (1, ∞) and g : (1, ∞) → R defined by 1
f ( x) = 32 x + 1, g ( x) = log 3 ( x − 1) are inverses.
2
Discrete mathematics 3e, page 84, problem 23
21) Let S={1,2,3,4,5} and f,g,h: S → S be functions defined by
f={(1,2),(2,1),(3,4),(4,5),(5,3)}
g={(1,3),(2,5),(3,1),(4,2),(5,4)}
h={(1,2),(2,2),(3,4),(4,3),(5,1)}
a) Find f o g ( x), g o f . Are these functions equal?
b) Explain why f and g have inverses but h does not. Find f −1 and g −1
22) Let S={1,2,3,4,5}, and T={1,2,3,8,9}, and define functions f : S →T and g : S → S
f={(1,8),(3,9),(4,3),(2,1),(5,2)} and g={(1,2),(3,1),(2,2),(4,3),(5,2)}.
e) Find I(3) for function f.
fof is not defined so I(3) not defined.
23) Use mathematical induction to prove the truth of each of the following assertions for all
n ≥1
8n − 3n is divisible by 5
52 n − 25 n is divisible by 7
10n +1 + 10n + 1 is divisible by 3
12 + 32 + 52 + K + (2n − 1) 2 =
n ( 2n − 1)( 2n + 1)
3
1.2.3 + 2.3.4 + 3.4.5 + ... + n(n + 1)(n + 2) =
n ( n + 1)( n + 2 )( n + 3)
4
Prove that 2 + 22 + 23 + . . . +2n = 2n+1 – 2 for all n ∈ N .
2
2
2
Q) Given 1 + 2 + K + n =
n ( n + 1)( 2n + 1)
, n ≥1
6
Find the sum of all the squares of the integers from 1 to 100(inclusive) that are non‐multiples of 5 (
= 100 × 101× 201 − 52 + 10 2 + 152 + ... + 100 2
)
= 2030100 − 52 (12 + 2 2 + 32 + ... + 20 2 )
= 2030100 − 52 (20 × 21× 41)
= 2030100 − 430500
= 1599600
24) The worst hockey team is allowed to draft one player from one of the top 3 teams. The
top 3 teams have 20, 24, 22 players respectively.
a) How many different possible players can the worst team pick?
Solution: 20 + 24 + 22 = 66
b) Suppose the worst team can pick one player from each of the top 3 teams? How many
different ways are there to pick the 3 players?
Solution: 20 · 24 · 22
25) How many license plates are available if each plate contains a sequence of 3 letters
followed by 3 digits?
26 · 26 · 26 · 10 · 10 · 10 = 17, 576, 000
26) (a) How many bit strings of length 8 are possible?
(b) How many of these start with a 1 or end with 00?
(c) How many have at most two 0s?
Answers: (a) 256 (b) 160 (c) 37
27) How many functions are there from a set with 6 elements to a set with 4 elements.
Answer: 4096
28) How many one-to-one functions are there from a set with 6 elements to a set with
(a) 4 elements (b) 6 elements (c) 10 elements. Answers: (a) 0 (b) 720 (c) 151 200
28. Each user on a computer system has a password, which is six to eight characters long, where each
character is a letter (case sensitive) or a digit. If each password must contain at least one digit and at
least one letter, how many possible passwords are there?
Answer: 167 410 838 583 040 =1167 1014