1 Set Builder Notation
Set builder notation is a precise way of expressing a set of objects in mathematics. When we
use set builder notation we are either encoding or decoding mathematics. As you saw in the
lecture notes there is a common format for set builder notation, however there is a degree of
latitude and freedom with this notation! Here, let’s go through a few decoding and encoding
exercises. This will help you complete Question 12 in the homework and help you learn
how to answer the question: “what the heck does he want!”
1.1 Decoding
In the following, we are going to translate into words what the set describes.
1. S={x∈R|x > 0}. This translates as the set Scontains all real numbers xsuch
that xis greater than zero. Let me show this parsed:
S = x∈R|x > 0
Set Scontains or equals all real numbers xsuch that xis greater than zero.
2. S={(x, x3)∈R2}. This translates as the set Scontains all points of the form (x, x3).
In other words it is the curve y=x3.Let me show this parsed:
S = (x, x3)∈R2
Set Scontains or equals all points of the form (x, x3) in two dimensions.
3. S={(x, y)∈R2|y=x3}. This is actually the same set of points as in the previous
one! Let’s look at the translation:
S = (x, y )∈R2|y=x3
Set Scontains all points in two dimensions such that y=x3.
This means there is not a unique way of writing down all sets. This is similar to any
language. There are many ways to say: “I need to work on mathematics”. We could
say, “I need to study mathematics” and other variations. Let me do one more decoding
example prior to working on some encoding examples.
4. S={(x, y, z)∈R2|z=x2+y2}. This set describes a paraboloid, that is, a bowl.
Let’s look at the translation:
S = (x, y, z)∈R3|z=x2+y2
Set Scontains all points in three dimensions such that the z coordinate is x2+y2.
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