Lecture 2+3

Question 1

What is an equivalence relation?

Answer 1

An equivalence relation means that two sets are the same, to show this the sets must show reflexivity(the set is equivalent to itself for all items in the set), symmetry(if set a is equivalent to set b then set b should be equivalent to set a) and transivity (if a is equivalent to b and b is equivalent to c then a is equivalent to c)

Question 2

what is an equivalence class?

Answer 2

Given an equivalence relation ∼ on A, we define the equivalence class [a] of a by:
[a] = {b|b ∼ a},
i.e. [a] is the subset of everything in A equivalent to a.
So essentially these are the different possible values in the set. n mod 2 would have two equivalence classes, 0 and 1.

Question 3

What is cardinality? How can we tell if two sets have the same cardinality?

Answer 3

The size of a set. Two sets have the same cardinality if they have a bijection, meaning that |X| is <= |Y| and |Y| <= |X|

Question 4

How can we tell if a set is uncountably infinite or infinite?

Answer 4

An infinite set with |A| = |natural numbers| is countably infinite. An infinite set in which this is not the case is uncountably infinite. To prove countably we find a bijection.

We can use Cantor’s diagonal argument to prove a set is uncountable. To do this we assume the set is countable and make a table to contain all the functions, we then find a way to make a new function that is not in the table, therefore our assumption that we could store all parts of the set in a table is wrong and the set must be uncountable.

Question 5

What does a recursive set definition involve?

Answer 5

A basis, a list of some objects which must be in the set.
A recursive step, how new objects are made from existing ones.
A closure rule, saying everything in X must be generated with a finite number of steps, normally unstated.

Question 6

Give the recursive definition for a sum

Answer 6

1. Basis: if n = 0 then m + n = m.
2. Recursive step: m + s(n) = s(m + n)
   Let’s try 3 + 2:
   s(s(s(0))) + s(s(0))
   =s(s(s(s(0))) + s(0))
   =s(s(s(s(s(0))) + 0))
   =s(s(s(s(s(0)))))

Question 7

What is induction?

Answer 7

A proof technique, we show that what we are trying to prove is true in the base case and then assume S is true for an element Xn, show it is true for Xn+1