1. Lecture 3 Strings and Things (Section 1.1)
Theory of Information Lecture 3
Theory of Information
Lecture 3 Strings and Things (Section 1.1)

2. Theory of Information Lecture 3
Congruency Modulo n
Theory of Information Lecture 3
Remember that Zn={0,1,…,n-1}
If  and  are integers, the following three conditions are equivalent:
1.  and  have the same remainder when divided by n
2. - is divisible by n
3. there is an integer k for which =+kn
When any (and hence all) of these conditions hold, we say that
and  are congruent modulo n, and write
(mod n)

3. Examples of Congruency
Theory of Information Lecture 3
53(mod 2)?
14143(mod 2)?
14410(mod 5)?
1934124(mod 10)?
-515(mod 5)?
-78(mod 2)?
-5  -2  147101316(mod 3)?

4. Theory of Information Lecture 3
Residue
Theory of Information Lecture 3
Given an integer , there is exactly one element in Zn congruent to 
(modulo n). Such an integer is called the residue of  modulo n.
The residue of 25 modulo 7 is
The residue of 32 modulo 7 is
The residue of 9925 modulo 2 is
The residue of 8764 modulo 2 is
Give me a number whose residue modulo 7 is 7

5. Theory of Information Lecture 3
Modular Arithmetic
Theory of Information Lecture 3
Let a,bZn. It, together with the following operations, is called
integers modulo n:
a+nb= the residue of a+b modulo n
anb= the residue of ab modulo n
Z5={0,1,2,3,4}. Let us just write + and  instead of +5 and 5 .
1+3=
1+4=
2+3=
4+4=
23=
44=
04=
14=
0+3=

6. Arithmetic Tables for Z2
Theory of Information Lecture 3
Multiplication modulo 2: 1
Addition modulo 2: 1

7. Theory of Information Lecture 3
Fields
Theory of Information Lecture 3
DEFINITION A field is a nonempty set F, together with two binary
operations on F, called addition (+) and multiplication (, often omitted),
satisfying the following properties:
Associativity: +(+)=(+)+; ()=().
Commutativity: +=+; =.
Distributivity: (+)= +.
Properties of 0 and 1: There are two distinct () elements: 0 (zero)
and 1 (one) such that, for all F, we have 0+= and 1=.
Inverse properties:
a) For every F, there is another element of F denoted by - and
called the negative of , such that +(-)=0;
b) For every nonzero F, there is another element of F denoted by
-1 and called the inverse of , such that (-1)=1.
With ordinary addition and multiplication, are the following sets fields?
{0,1,…} {…-2,-1,0,1,2,…} The set of rational numbers

8. Integers Modulo 2 Is a Field
Theory of Information Lecture 3
For Z2={0,1} (as well as all other Zn), associativity, commutativity and
distributivity hold because they hold for the ordinary + and .
What is the negative of: 0? 1?
What is the inverse of: 1?
Subtraction - means +(-).
In our case,
0-0=
0-1=
1-0=
1-1=

9. Integers Modulo 3 Is a Field
Theory of Information Lecture 3
What is the negative of: 0? 1? 2?
What is the inverse of: 1? 2?

10. Integers Modulo 4 Is not a Field
Theory of Information Lecture 3
Z2={0,1,2,3}
What would be the inverse of 2?
20=
21=
22=
23=

11. Theory of Information Lecture 3
When is Zn a field?
Theory of Information Lecture 3
An integer n is said to be prime iff n has no divisors other than 1 and n.
Prime numbers: 2,3,7,11,13,17,19,23,29,31,41,…
THEOREM The set Zn of integers modulo n is a field if and
only if n is a prime number.

12. Theory of Information Lecture 3
Strings
Theory of Information Lecture 3
An alphabet is a finite, nonempty set S={s1,…,sn} of symbols.
A string, or word over S is a finite sequence of elements of S.
E.g., when S={a,b,…,z}, then
school, bbb, a, xyz
are strings/words over S. So is the empty string  (theta).
Stings over the alphabet {0,1} are called binary strings, and the
symbols 0 and 1 are called bits.
The complement of a binary string x is the result of replacing in x every
0 by 1 and every 1 by 0. Denoted by xc.
Strings over {0,1,2} are called ternary strings.

13. Theory of Information Lecture 3
Strings
Theory of Information Lecture 3
The juxtaposition (concatenation) of strings x and y is xy.
len(x) denotes the length of string x.
len(0010)=4, len(0)=1, len()=0.
When S is an alphabet:
S* stands for the set of all strings over S
Sn stands for the set of all strings over S whose length is n
Sn stands for the set of all strings over S whose length is n
THEOREM Let S be an alphabet of size k.
|Sn|=kn
|Sn|= 1+k+k2+…+kn

14. Theory of Information Lecture 3
Strings
Theory of Information Lecture 3
THEOREM 1.1.4
In (Z2)n, the number of strings with exactly k 0s is ( )
In (Zr)n, the number of strings with exactly k 0s is ( )(r-1)n-k n k n k
Proof. Part 1: There is one string with exactly k 0s for every way of
choosing k of the n positions in which to place 0s. Part 2: For each
choice of where to place 0s, there are (r-1)n-k ways to arrange the
remaining (r-1) symbols in the remaining (n-k) positions.

15. Theory of Information Lecture 3
Homework
Theory of Information Lecture 3
Example of the textbook.
Exercises 1,3,4,5,7,8 of Section 1.1 of the textbook.