Study unit 3 Sets COS1501/1 36Examples From “x > 3” we get the specific statement “5 > 3” if we push the name 5 into the space kept open by the variable x. Another example: Suppose we want to talk about the set of all positive integers less than 5. Using set-builder notation, we can write {x | x is a positive integer less than 5}. Abbreviations: Set membership Suppose we want to say “3 is a memberof Z”. Then it will be convenient if we can abbreviate the phrase “is a member of”. The symbol we usually use for this abbreviation is “∈”. So, “3 is a member of Z” can be written as “3 ∈Z”. We can also say “3 is an elementof Z”. It could be that some element, say −2, is not a member of the set Z+, then we say “−2 is not a memberofZ+”, and we may write “−2 ∉Z+”. Similarly, in order to say “2 is a member of {0, 1, 2}”, we may write “2 ∈{0, 1, 2}”, and to say “3 is not a member of {0, 1, 2}”, we may write “3 ∉{0, 1, 2}”.The symbol “∈” is a streamlined version of the letter epsilon (ε) which is the Greek version of the English letter e, and e is the first letter of the word “element”. Sets should be well-defined. In general, a set can be described in different ways, and we illustrate this by the following examples: Examples The set of even non-negative integers less than 10 can be described in (at least) three ways: {0, 2, 4, 6, 8}, {x | x is an even non-negative integer less than 10}, and {x | x ∈Z≥, x is an even integer less than or equal to 8}. These three descriptions might look different, but clearly they refer to the same collection of things. We indicate this by writing {0, 2, 4, 6, 8} = {x | x is an even non-negative integer less than 10} = {x | x ∈Z≥, x is an even integer less than or equal to 8}. Another example: Suppose we want to talk about the set of all negative integers greater than −
5. We can describe this set by using
list notation: {
−
4,
−
3,
−
2,
−
1}, or
set-builder notation: {x | x is a negative integer greater than
−
5}.
We may write
{
−
4,
−
3,
−
2,
−
1} = {x | x is a negative integer greater than
−
5}
= {y | y
∈
Z
,
−
4
≤
y < 0}.
There are also other alternatives that can describe this set.

Study unit 3
Sets
COS1501/1
37
Here the “=” stands for “is the same set as” or, if you prefer, “is equal to”.
Note that “x is an even non-negative integer less than 10”, and “x is a negative
integer greater than
−
5” are referred to as property descriptions.
Note: Different variables can be used when defining a set,
e.g. {y | y
∈
Z
,
−
4
≤
y < 0} = {z | z
∈
Z
,
−
4
≤
z < 0}, and other alternatives can also
define this set.
But how do we check whether or not two sets are equal? We can’t
always expect it to be obvious.
Example
Consider, on the one hand
{x | x is a real number and 1 < x < 2}, and on the other hand
{x | x is a real number and x
2
−
3x + 2 < 0}.

#### You've reached the end of your free preview.

Want to read all 188 pages?

- Spring '14
- Addition, Negative and non-negative numbers, Natural number, Rational number