*Things* have *Attributes*

All the *Things* that share a given *Attribute* are a **Set**

For example:

- red things
- wild flowers
- smart students

April 24th, 2017

*Things* have *Attributes*

All the *Things* that share a given *Attribute* are a **Set**

For example:

- red things
- wild flowers
- smart students

When we say “all the *things*”, what do we mean by “all”?

In principle *all* can mean literally everything, real or imaginary

In practice we mean *all thing of some class*. This is called “the **Universe**”

For example:

- In
*“red things”*the universe can be “physical objects with color” - In
*"wild flowers*" the universe can be “all flowers” - In
*"smart students*" the universe can be “all students”

If it is not clear from the context, we have to define explicitly the universe

For notation we often use \(U\). That is

\(U\) is the set of all things that are relevant to the current issue

If a *thing* \(x\) has an *attribute* \(A\), then the predicate \(A(x)\) is true

Now we will use the same symbol \(A\) to represent the set of all things with that property

We say that \(A\) is the *set* of all things \(x\) such that \(A(x)\) is true \[A=\{x : A(x)\}\]

Now if \(x\) has the property \(A\), we can say that \(x\in A\)

**Notice:** With parenthesis it is a logic phrase, without it is a set

We said that *attributes* only exist when some *thing* has the attribute

In more general we can imagine attributes that can exist, but there is nothing in the universe that has that attribute

(like “being a time machine”)

If an *Set* does not have anything, we say it is an **empty set**

We write \(\emptyset\), meaning “the empty set”

What about the things that *do not* have an attribute?

We can also define the set of things \(x\) such that \(\neg A(x)\)

We call this set the **complement** of the set \(A\)

We write it as \(\overline{A}\) or as \(A^C\), so

\[\overline{A} = \{x: \neg A(x)\}\] which can also be written \[\overline{A} = \{x: x\not\in A\}\]

We can give names to all sets that we make from Logical predicates

For example, the set of all \(x\) with property \(A\) AND property \(B\) is called the **intersection** of the set \(A\) and the set \(B\)

\[A\cap B=\{x: x\in A\, \&\, x\in B\} =\{x: A(x)\, \&\, B(x)\}\]

Therefore, if we know that \(x\in A\cap B\) then we know that \(x\in A\) AND \(x\in B\)

**Notice:** The *intersection* of two roads is the part that belongs to both roads

In the same way we can define the set of all \(x\) with property \(A\) OR property \(B\)

It is called the **union** of the sets \(A\) and \(B\)

\[A\cup B = \{x: x\in A\, \vert\, x\in B\} =\{x: A(x)\, \vert\, B(x)\}\]

Therefore, if we know that \(x\in A\cup B\) then we know that \(x\in A\) OR \(x\in B\)

What about all the elements of \(A\) that are not elements of \(B\)?

The **set difference** of \(A\) and \(B\) is the set of all elements in \(A\) that are not elements of \(B\)

\[A\setminus B = \{x: x\in A \,\&\, x\not\in B\}\]

We can draw two sets and identify all the elements we just described

- \(U\)
- \(\overline{A}\)
- \(A\cap B\)
- \(A \cup B\)
- \(A\setminus B\)