Logic notation, baby set theory and countability
Language not only helps us communicate thoughts but also shapes them and inuences the way
we perceive the world. It is therefore no surprise that it is a very good way to communic
The trouble with naive set theory
Sets are used throughout mathematics. Typically they dened in terms of properties:
S = cfw_x : x satises property p.
This is a very sensible way to go about constructing sets and works very well for a larg
Axioms and operations
Recap of our aim
We start introducing the axioms that we will use to build sets. The typical axiomatic approach
is to mathematics is to begin by listing the axioms and then carefully deducing propositions
about the entiti
Relations and functions
The material in this chapter has largely been covered in Chapter I. In this second presentation
we emphasise the foundational aspects: how relations and functions can be dened from the
axioms introduced so far. We will al
In this chapter we will construct the natural numbers using sets. we will then dene addition,
multiplication as set operations and note they have the properties we expect.
To create some perspective it is good to know how numbers
MTH 246 - Midterm 1 practice solutions
Baby set theory questions
A set A is countable if there exists a surjection f : N A.
(i) The set A = cfw_2z : z Z of even integers is countable as there exists a bijection
f : N A given by
, n is even
MTH 246 - Midterm 2 practice solutions
There will be four questions of equal weight.
Question 1. Dene the following terms:
- successor of a set a: a+ = a cfw_a.
- an inductive set: a set T is inductive, if T and a T = a+ T.
- a natural number: a set belon
MTH 246 - Homework 2 solutions
Question 1. You are given three formulae. Identify the variables, indicate which variables are
free, identify all the atomic formulae and write down two non-atomic sub-formulae, that is
formulae that appear in the given form
MTH 246 - Homework 1 solutions
Question 1. Use logic notation to write down formal denitions for the following propositions.
You are only allowed to use the symbols , , , , = , . Functions should be treated
as subsets of Cartesian products and the notatio
MTH 246 - Homework 3 solutions
Question 1. Let A be a set. Prove that P (A) A. Notice that this proves that there does
not exist a set of all sets. [Hint: adapt the proof of Theorem 2A]
We argue by contradiction. Suppose that P (A) A and consider the set
MTH 246 - Homework 5 solutions
Question 1. Before attempting this question you are advised to read through the example of
innite Cartesian products given on pp. 89 of Chapter 3 of the on-line notes.
Let S and set AS = \ S be its complement. The question f
MTH 246 - Homework 4 Solutions
Question 1. Quick questions no need to oer lengthy explanation.
(i) Given x R let Ax = cfw_y R : y x. What is
It is equal to R. A justication was not required but we include it. Ax R for all x R and
so xR Ax R. For
MTH 246 - Homework 6 solutions
Question 1. Recall that given a function F : B B and a subset S B we say that S is
closed under F when F [S ] S. In this question we construct (in two dierent ways) the closure
of a subset A B , which should be thought of as
MTH 246 - Homework 8
Due by 4pm Thursday November 14
Each question is worth 10pts.
Question 1. (i) Prove that addition in R is commutative and associative. You may use the
fact that addition in Q is associative and commutative.
Let x, y and z be real num
MTH 246 - Homework 7 solutions
Due by 4pm Thursday November 7
Each question is worth 12pts
Question 1. (i) Let A = and F : A A A be the function given by
m, n , p, q
) = m + p, n + q .
Here + stands for addition in . Prove that F is compatible with the