The University of Sydney
MOOC Introduction to Calculus
Notes for ‘Real line, decimals and significant figures’
Important Ideas and Useful Facts:
(i) Sets and elements: A set is a collection of objects, referred to as elements. A set may
be represented, for example, by a list of elements surrounded by curly brackets and
separated by commas, or using set builder notation {. . . | . . .}, where the vertical line is
an abbreviation for “such that”. For example, { x | x is a natural number less than 5 }
and {0, 1, 2, 3, 4} represent the same set, whose elements are precisely 0, 1, 2, 3 and 4.
(A natural number is a whole counting number, including zero. Note that some people
do not count zero as a counting number, but we do in this course!)
(ii) Element symbol: The symbol ∈ is an abbreviation for “is an element of”, and 6∈ is an
abbreviation for “is not an element of”. For example, if
A = { x | x is a natural number less than 5 } ,
then 2 ∈ A, but 5 6∈ A.
(iii) Subset symbols: If A and B are sets and we write A ⊆ B or B ⊇ A , then we mean that
every element of A is also an element of B, and say that A is a subset of B. For example
{1, 2, 3} ⊆ {1, 2, 3, 4} and {1, 2, 3, 4} ⊇ {1, 2, 3}, but {1, 2, 3, 4} 6⊆ {1, 2, 3}.
(iv) Equality of sets: If A and B are sets then A = B if and only if A ⊆ B and B ⊆ A, that
is, A and B have precisely the same elements. Order and repetition are not important.
For example, {1, 2, 3, 4} = {4, 1, 3, 2} = {4, 1, 3, 1, 2, 3}.
(v) Intersection, union and slash: If A and B are sets then put
(a) A ∩ B = { x | x ∈ A and x ∈ B }, called the intersection of A and B.
(b) A ∪ B = { x | x ∈ A or x ∈ B }, called the union of A and B.
(c) A\B = { x | x ∈ A and x 6∈ B }, called A slash B, the result of removing from A all
elements from B.
(vi) Natural numbers: The set N = {0, 1, 2, 3, . . .} of natural numbers forms a number system,
closed under addition and multiplication, by which we mean that if m and n are natural
numbers, then m + n and mn (the result of multiplying m by n) are also natural numbers.
(vii) Integers: The set Z = {0, ±1, ±2, ±3, . . .} of integers forms a number system, closed under
addition, subtraction and multiplication.
(viii) Rationals: The set Q = {a/b | a, b ∈ Z, b 6= 0} of fractions, also called rational numbers
(derived from the word ratio), forms a number system, closed under addition, subtraction,
multiplication and division by nonzero elements. To add and multiply rational numbers,
use the rules
a c ad + bc a c ac
+ = and · = .
b d bd b d bd
1
MOOC Introduction to Calculus
Notes for ‘Real line, decimals and significant figures’
Important Ideas and Useful Facts:
(i) Sets and elements: A set is a collection of objects, referred to as elements. A set may
be represented, for example, by a list of elements surrounded by curly brackets and
separated by commas, or using set builder notation {. . . | . . .}, where the vertical line is
an abbreviation for “such that”. For example, { x | x is a natural number less than 5 }
and {0, 1, 2, 3, 4} represent the same set, whose elements are precisely 0, 1, 2, 3 and 4.
(A natural number is a whole counting number, including zero. Note that some people
do not count zero as a counting number, but we do in this course!)
(ii) Element symbol: The symbol ∈ is an abbreviation for “is an element of”, and 6∈ is an
abbreviation for “is not an element of”. For example, if
A = { x | x is a natural number less than 5 } ,
then 2 ∈ A, but 5 6∈ A.
(iii) Subset symbols: If A and B are sets and we write A ⊆ B or B ⊇ A , then we mean that
every element of A is also an element of B, and say that A is a subset of B. For example
{1, 2, 3} ⊆ {1, 2, 3, 4} and {1, 2, 3, 4} ⊇ {1, 2, 3}, but {1, 2, 3, 4} 6⊆ {1, 2, 3}.
(iv) Equality of sets: If A and B are sets then A = B if and only if A ⊆ B and B ⊆ A, that
is, A and B have precisely the same elements. Order and repetition are not important.
For example, {1, 2, 3, 4} = {4, 1, 3, 2} = {4, 1, 3, 1, 2, 3}.
(v) Intersection, union and slash: If A and B are sets then put
(a) A ∩ B = { x | x ∈ A and x ∈ B }, called the intersection of A and B.
(b) A ∪ B = { x | x ∈ A or x ∈ B }, called the union of A and B.
(c) A\B = { x | x ∈ A and x 6∈ B }, called A slash B, the result of removing from A all
elements from B.
(vi) Natural numbers: The set N = {0, 1, 2, 3, . . .} of natural numbers forms a number system,
closed under addition and multiplication, by which we mean that if m and n are natural
numbers, then m + n and mn (the result of multiplying m by n) are also natural numbers.
(vii) Integers: The set Z = {0, ±1, ±2, ±3, . . .} of integers forms a number system, closed under
addition, subtraction and multiplication.
(viii) Rationals: The set Q = {a/b | a, b ∈ Z, b 6= 0} of fractions, also called rational numbers
(derived from the word ratio), forms a number system, closed under addition, subtraction,
multiplication and division by nonzero elements. To add and multiply rational numbers,
use the rules
a c ad + bc a c ac
+ = and · = .
b d bd b d bd
1
