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Sets for Advanced Mathematics
- Institution
- University Of The Freestate (UFS)
These are simple notes that will help you understand Sets quicker than most.
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MeaningThis instructional document serves as a tool in aiding first year students in understanding Sets.
What are sets?
A set is a mathematical model for a collection of different things. They contain elements which can
be mathematical objects of any kind, such as numbers, symbols, points in space, other sets, etc…
Consider set J, where
J
{15, 16}
8 4 ϕ4
6 976 1
In roster
notation The numbers inside the
(which will curly brackets {} separated
be used a lot) by a coma, are the
J = { 1, 4, 6, 8, 976, {15, 16}, ϕ } ➔ elements. Sometimes you
will see ‘…’ which imply that
Also, notice that one of the elements in J is a the set extends to infinity,
set. e.g., B = {…, -1, 2, 5, 8, …}.
ϕ is an empty set or {}.
Repetition is not important, {1,1,1} = {1}.
What is the Cardinality of set J?
Cardinality refers to the number of elements in a set. So set J has a cardinality of 7.
Describing a large set.
Take G = { 𝑥 𝜖 ℤ | 2 < 𝑥 < 101 } , which translates to, ‘x is an element of integers, where ( | ) x is
greater than 2 and less than 101. So, G = { 3, 4, 5, 6, …, 98, 99, 100 }.
Other large sets you can look up: Natural number ℕ, integers ℤ, rational number ℚ, real numbers ℝ,
and complex numbers ℂ.
Describing a set using another set.
From ‘Describing a large set’, ℤ is an infinitely large set of all integers (from negative to positive
infinity). So, we described D using the set ℤ. You can also use D to describe another set, e.g. D
2
= { 𝑥 𝜖 𝐺 | 𝑥 < 37 } = { 9, 16, 25, 36 }.
Equal Sets
Say set H = { 36, 36, 9, 9, 9, 16, 25, 25, 25, 25 } = { 9, 16, 25, 36 }, then we can also say the D = H, where
D is the set from ‘Describing a set using another set’.
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