MA251 Set Theory
Assigned Problems (No. 1)
SOLUTIONS
1. The following story was told by a Chinese philosopher Han Fei Zi (around 281 - 233
BC).
An armorer of the State of Chu boldly claimed to make the best spears
and shields.
"My shields are so strong; t
Math 215A,B
Set Theory (Page 1 of 2)
Fall term, 2012
Sections
Section Instructor
Lecture time
Lect room
Oce Hours
A
Shengda Hu MWF 12:30 13:20 DAWB 2106
MF 13:30 14:30, Tu 9:20 10:00
B
Craig Sloss MWF 9:30 10:20
BA202
MWF 10:30 11:30, TuTh 10:00 11:30
Ins
Real numbers
Example 1. The bounded subset S = fx : x2 < 2g of Q has no least upper
bound in Q. Therefore the linearly ordered set Q is not order-complete.
In the ancient Greece, mathematicians tried to represent all rational number
geometrically on a str
MA251 Set Theory
Assigned Problems (No. 3)
1. Let B n P1, 2, , n. The set
0, 1 n a 1 , a 2 , , a n : a i 0, 1
is called the set of binary sequences of length n.
(a) Describe a function f : 0, 1 n B n which is a bijection, and give its
inverse.
Solution. D
Assignment 5 Solutions
Corrections:
page 1: Line
4: ". we can take p 2 L & p0 2 R such that "
page 3: Line 3: ". with
< p0 <
+
1
&
< q0 <
+ , and hence"
The Definition of Real Numbers Using Dedekind Cuts
Example 1. The bounded subset S x : x 2 2 of Q has no least upper
bound in Q. Therefore the linearly ordered set Q is not order-complete.
In the ancient Greece, mathematicians tried to represent all ratio
MA251 Set Theory
Assigned Problems (No. 3)
1. Let B n P1, 2, , n. The set
0, 1 n a 1 , a 2 , , a n : a i 0, 1
is called the set of binary sequences of length n.
(a) Describe a function f : 0, 1 n B n which is a bijection, and give its
inverse.
Solution. D
PARTITIONS
Let S be a nonempty set. A partition of S is a subdivision of S into nonoverlapping, nonempty
subsets. Precisely, a partition of S is a collection P = {21,-} of nonempty subsets of S such that
(i) Each is E S belongs to one of the A f.
(it) T
MA251 Set Theory
Assigned Problems (No. 3)
1. Let B n P1, 2, , n. The set
0, 1 n a 1 , a 2 , , a n : a i 0, 1
is called the set of binary sequences of length n.
(a) Describe a function f : 0, 1 n B n which is a bijection, and give its
inverse.
(b) Use par
MA251 Set Theory
Assigned Problems (No. 2)
1. Let R1 be the relation dened on Z, the set of all integers, by aR1 b if
a + b is even, and let R2 be the relation dened on Z by aR2 b if a + b is odd.
Determine whether they are equivalence relations.
2. A rel