Question 1 (10 marks)
(i) Write the statement Each MATH121 student will pass at least one MATH121 assignment
using quantiers. Do you think that this statement is true?
(ii) Write the statement x R, y R, x < y in English (that is, without mathematical
symb
Lecture 11
Relations and Functions 3
This material is covered in Section 5.5 of the course notes.
Recap (Lecture 10)
Suppose A is a set and R is an equivalence relation on A. For
each a A, the equivalence class of a, denoted [a], is the
set [a] = cfw_x A
Lecture 8
Set Theory 2
This material is covered in Sections 4.4.14.6 of the course notes.
Recap (Lecture 7)
A set is just a collection of objects. The objects are called
elements.
The empty set is the set with no elements.
Subset: A B if and only if every
Lecture 9
Relations and Functions 1
This material is covered in Sections 5.15.3 of the course notes.
The Cartesian Product
Denition. Let A and B be sets, and let a A and b B . An
ordered pair (a, b ) is a pair of elements with the property that
(a, b ) =
Lecture 10
Relations and Functions 2
This material is covered in Section 5.4 of the course notes.
Recap (Lecture 9)
If A and B are sets, then the Cartesian product of A and B ,
denoted A B , is A B = cfw_(a, b ) : a A b B .
If (x , y ) R , then we write x
Lecture 4
Predicate Logic
This material is covered in Section 2 of the course notes.
Introduction to Predicate Logic
In the last three lectures we discussed the logical analysis of
compound statements, which are statements made up of simple
statements joi
Question 1 (12 marks)
(i) Prove that if a, b Z and a | b or a | c, then a | bc.
(ii) Prove that the product of an odd integer and an even integer is even.
(iii) Write down all the positive divisors of 75 and all the positive divisors of 18.
(iv) Using (ii
Question 1 (10 marks)
(i) Write the statement Each MATH121 student will pass at least one MATH121 assignment
using quantiers. Do you think that this statement is true?
(ii) Write the statement x R, y R, x < y in English (that is, without mathematical
symb
Lecture 5
Methods of Proof 1
This material is covered in Sections 3.13.2.2 of the course notes.
Logic Recap
Let P and Q be statement variables. Then
P
means
P Q
means
P Q
means
P = Q
means
P Q
means
Logic Recap ctd
Question: Is the statement
(P = Q ) Q =
Lecture 1
Logic 1
This material is covered in Sections 1.11.2.3 of the course notes.
Introduction to logic
Arithmetic is the study of numbers and the operations addition,
subtraction, multiplication and division we can apply to numbers
to form new numbers
Lecture 6
Methods of Proof 2
This material is covered in Sections 3.2.33.5 of the course notes.
Recap (Lecture 5)
An argument is a sequence of statements. All the statements
in an argument apart from the last are called assumptions; the
last statement in
Lecture 2
Logic 2
This material is covered in Sections 1.2.41.3.2 of the course notes.
Recap (Lecture 1)
A statement is a sentence that is true or false but not both.
New statements are formed from old statements by use of
logical connectives:
In words
no
Lecture 3
Logic 3
This material is covered in Sections 1.3.31.4.2 of the course notes.
Recap (Lecture 2)
The conditional of Q by P is If P , then Q or P implies Q
and is denoted P = Q .
The biconditional of P and Q is P if and only if Q and is
denoted P
Lecture 7
Set Theory 1
This material is covered in Sections 4.14.4 of the course notes.
Introduction to sets
The words set and element are undened terms of set theory
just like the terms sentence, true and false are undened
terms of logic. But we think of