Set Theory

Definitions

• What is a set - A collection of distinct element.
• What is a element - Is a member of a set
• Notation - Usually denoted using {} e.g. for numbers {0,1,2,3,5,8}
• Cardinality - number of elements in the set. Above set would be 6

Type of Sets

Countably Infinite sets

Each element can be place in a one-to-one correspondence with natural numbers, or You can line up every element so each one has its own unique index N - Natural numbers (Positive integers)

{0,1,2,3,...}
{1,2,3,4...}

Z - Integers

{0,1,-1,2,-2,3,-3,...}

Z+ - Integers

{1,2,3,4,...}

Q - Rational numbers (p/q) where p is an integer and q is a non-zero integer

Uncountably Infinite sets

"More" elements than there are natural numbers. e.g. the numbers between 0 and 1, there is 0.1, 0.11, 0.111 or 0.2, 0.22, 0.222 etc R - Real Numbers C - Complex Numbers

Finite sets

A set with a limited number of elements. They have a cardinality of a natural number. E.g.

• primary colours (red, yellow, blue).
• days of the week
• hours in the day

And this is why I am here
Set of remainders of integers modulo n where n = integer. So it is asking how many different integer remainders are there for n which is {0,1,2,3,4}. So for integer modulo n

• Cardinality = n
• Largest number in the set is n-1

So we are focusing on the remainder, the amount left over, not the Quotient or the number of times it is divisible by. For example

(7+3)mod5
10 is divisible by 5 exactly two times
2 is the quotient
0 is the remainder

So the answer is 0