Test of Mathematics for University Admission
Notes on Logic and Proof
© UAT‐UK 2024 Version 1.3 May 2024
, Notes for Paper 2 of
The Test of Mathematics for University Admission
You should check the website regularly for updates to these notes. You can tell if
there have been any updates by looking at the date and version number on the front
cover.
Introduction
The formal side of mathematics ‐ that of theorems and proofs ‐ is a major part of the
subject and is the main focus of Paper 2. These notes are intended to be a brief
introduction to the ideas involved, for the benefit of candidates who have not yet
met them within their mathematics classes or within their wider mathematical
reading.
Mathematics, in part, is about working out the relationships between
(mathematical) statements. And it is very important that everyone (that is, all
mathematicians) write and talk about these relationships in the same standard way.
It is important because mathematics is expressed formally using a rigorous language
and if different people mean different things when they use the same words, then it
would be difficult to ensure that everyone is talking about (and agreeing about) the
very same things. Learning the basics of the rules and terms used by mathematicians
is essential if you are to be able to understand and contribute to mathematics. These
notes are written to help introduce you to some of the terms used regularly by
mathematicians – specifically the terms we have included in Section 2 of the TMUA
specification.
Before you launch into reading through what we have written, there are a few things
to keep in mind:
1. This guide is designed to be a brief overview. It is not designed to be an
extensive textbook, but it should be enough to allow you to get a good
understanding of the topics in Section 2 of the specification. It should also be
sufficient to enable you to understand and tackle the questions we will ask in
the admissions test.
2. As you read through the guide, make sure you take the time to think through
everything very carefully. Many of the ideas set out here are quite subtle and
take some time to grasp, so skimming through everything is certainly not
enough to ensure you have a good understanding. Rather, you should play
around with the ideas as you meet them, and try to come up with your own
examples. In other words, read through things actively with a pencil and
paper to hand; think carefully about everything, draw your own pictures,
write out your own examples, and so on.
3. Another thing you should do is to try to read more widely on the topics set
out here. The internet has some good explanations and examples of the ideas
, we outline and there are some good books available from libraries that might
also help.
4. A good way to test if you have understood something is to see if you would
be able to explain the ideas to a friend or a class of students. If you get a
chance, it is always useful to study the ideas and talk about them with other
people – again, maybe with others in your maths classes or with your maths
teacher.
5. As you work through these notes, make sure you are aware that the language
used by mathematicians has very precise meanings and these do not always
coincide with the way words are used in everyday casual contexts. For
instance, if I am told that I may have “jelly or cake” for pudding, I would
probably assume it meant I could have jelly or cake but not both. However, in
mathematics, ‘or’ means ‘one, or the other, or both’, so if I were being
offered pudding by a mathematician then I could have both jelly and cake if I
wanted.
6. Throughout, we have tried to explain the ideas in at least two different ways:
one using the formal notion of truth tables and one using a more intuitive
diagrammatic approach. It is worth making sure you understand both
approaches and how they relate to each other.
7. Finally, a small note of caution: in this specification we deliberately adopt a
simple and slightly naïve view of the ideas we are trying to explain rather
than one that sets out all the deep subtleties that abound in mathematics
and the philosophy of logic/mathematics. We have been as rigorous as
necessary to achieve our aims, but you should not take what we have written
as the perfect and final word on things and we have deliberately avoided
some issues as they would only complicate matters unnecessarily. For
instance, some mathematicians or philosophers might take issue with our
examples of statements or our notion of truth and so on. For what we aim to
achieve, these issues are not relevant, but that is not to say they aren’t
interesting.
, The relevant part of the specification
SECTION 2
This section sets out the scope of Paper 2. Paper 2 tests the candidate’s ability to
think mathematically: the paper will focus on testing the candidate’s ability to
understand, and construct, mathematical arguments in a variety of contexts. It will
draw on the mathematical knowledge outlined in SECTION 1 in the test specification.
The Logic of Arguments
Arg1 Understand and be able to use mathematical logic in simple situations:
The terms true and false;
The terms and, or (meaning inclusive or), not;
Statements of the form:
if A then B
A if B
A only if B
A if and only if B
The converse of a statement;
The contrapositive of a statement;
The relationship between the truth of a statement and its converse
and its contrapositive.
Note: candidates will not be expected to recognise or use symbolic notation for any
of these terms, nor will they be expected to complete formal truth tables.
Arg2 Understand and use the terms necessary and sufficient.
Arg3 Understand and use the terms for all, for some (meaning for at least one),
and there exists.
Arg4 Be able to negate statements that use any of the above terms.
Mathematical Proof
Prf1 Follow a proof of the following types, and in simple cases know how to
construct such a proof:
Direct deductive proof (‘Since A, therefore B, therefore C, …,
therefore Z, which is what we wanted to prove.’);
Proof by cases (for example, by considering even and odd cases
separately);
Proof by contradiction;
Disproof by counterexample.
Prf2 Deduce implications from given statements.
Prf3 Make conjectures based on small cases, and then justify these conjectures.
Notes on Logic and Proof
© UAT‐UK 2024 Version 1.3 May 2024
, Notes for Paper 2 of
The Test of Mathematics for University Admission
You should check the website regularly for updates to these notes. You can tell if
there have been any updates by looking at the date and version number on the front
cover.
Introduction
The formal side of mathematics ‐ that of theorems and proofs ‐ is a major part of the
subject and is the main focus of Paper 2. These notes are intended to be a brief
introduction to the ideas involved, for the benefit of candidates who have not yet
met them within their mathematics classes or within their wider mathematical
reading.
Mathematics, in part, is about working out the relationships between
(mathematical) statements. And it is very important that everyone (that is, all
mathematicians) write and talk about these relationships in the same standard way.
It is important because mathematics is expressed formally using a rigorous language
and if different people mean different things when they use the same words, then it
would be difficult to ensure that everyone is talking about (and agreeing about) the
very same things. Learning the basics of the rules and terms used by mathematicians
is essential if you are to be able to understand and contribute to mathematics. These
notes are written to help introduce you to some of the terms used regularly by
mathematicians – specifically the terms we have included in Section 2 of the TMUA
specification.
Before you launch into reading through what we have written, there are a few things
to keep in mind:
1. This guide is designed to be a brief overview. It is not designed to be an
extensive textbook, but it should be enough to allow you to get a good
understanding of the topics in Section 2 of the specification. It should also be
sufficient to enable you to understand and tackle the questions we will ask in
the admissions test.
2. As you read through the guide, make sure you take the time to think through
everything very carefully. Many of the ideas set out here are quite subtle and
take some time to grasp, so skimming through everything is certainly not
enough to ensure you have a good understanding. Rather, you should play
around with the ideas as you meet them, and try to come up with your own
examples. In other words, read through things actively with a pencil and
paper to hand; think carefully about everything, draw your own pictures,
write out your own examples, and so on.
3. Another thing you should do is to try to read more widely on the topics set
out here. The internet has some good explanations and examples of the ideas
, we outline and there are some good books available from libraries that might
also help.
4. A good way to test if you have understood something is to see if you would
be able to explain the ideas to a friend or a class of students. If you get a
chance, it is always useful to study the ideas and talk about them with other
people – again, maybe with others in your maths classes or with your maths
teacher.
5. As you work through these notes, make sure you are aware that the language
used by mathematicians has very precise meanings and these do not always
coincide with the way words are used in everyday casual contexts. For
instance, if I am told that I may have “jelly or cake” for pudding, I would
probably assume it meant I could have jelly or cake but not both. However, in
mathematics, ‘or’ means ‘one, or the other, or both’, so if I were being
offered pudding by a mathematician then I could have both jelly and cake if I
wanted.
6. Throughout, we have tried to explain the ideas in at least two different ways:
one using the formal notion of truth tables and one using a more intuitive
diagrammatic approach. It is worth making sure you understand both
approaches and how they relate to each other.
7. Finally, a small note of caution: in this specification we deliberately adopt a
simple and slightly naïve view of the ideas we are trying to explain rather
than one that sets out all the deep subtleties that abound in mathematics
and the philosophy of logic/mathematics. We have been as rigorous as
necessary to achieve our aims, but you should not take what we have written
as the perfect and final word on things and we have deliberately avoided
some issues as they would only complicate matters unnecessarily. For
instance, some mathematicians or philosophers might take issue with our
examples of statements or our notion of truth and so on. For what we aim to
achieve, these issues are not relevant, but that is not to say they aren’t
interesting.
, The relevant part of the specification
SECTION 2
This section sets out the scope of Paper 2. Paper 2 tests the candidate’s ability to
think mathematically: the paper will focus on testing the candidate’s ability to
understand, and construct, mathematical arguments in a variety of contexts. It will
draw on the mathematical knowledge outlined in SECTION 1 in the test specification.
The Logic of Arguments
Arg1 Understand and be able to use mathematical logic in simple situations:
The terms true and false;
The terms and, or (meaning inclusive or), not;
Statements of the form:
if A then B
A if B
A only if B
A if and only if B
The converse of a statement;
The contrapositive of a statement;
The relationship between the truth of a statement and its converse
and its contrapositive.
Note: candidates will not be expected to recognise or use symbolic notation for any
of these terms, nor will they be expected to complete formal truth tables.
Arg2 Understand and use the terms necessary and sufficient.
Arg3 Understand and use the terms for all, for some (meaning for at least one),
and there exists.
Arg4 Be able to negate statements that use any of the above terms.
Mathematical Proof
Prf1 Follow a proof of the following types, and in simple cases know how to
construct such a proof:
Direct deductive proof (‘Since A, therefore B, therefore C, …,
therefore Z, which is what we wanted to prove.’);
Proof by cases (for example, by considering even and odd cases
separately);
Proof by contradiction;
Disproof by counterexample.
Prf2 Deduce implications from given statements.
Prf3 Make conjectures based on small cases, and then justify these conjectures.
