What is true and what is not

We will deal with statements that are either True or False

For example

• The patient has an infection
• The antibiotic kills the bacteria
• It is raining
• I will win the lottery
• You will pass this course

Sometimes they are also called propositions

Simplification and Generalization

In math, when we want to make things general, we get rid of the “noise”, that is, the details that make things complicated and are not relevant to us.

In this case we are going to represent statements with uppercase letters

• \(A\), \(B\), \(C\), …
• T means True, F means False

Complex statements

We can change and modify statements to make new ones

We can say

• “the patient has an infection and the antibiotic kills the bacteria, but it is not raining”
• “You will pass this course or I will win the lottery”

Negation

The simplest way to change a statement is to negate it

For instance, if \[A\] means “It is raining”, then \[\text{not }A\] means “it is not raining”

Combining two statements

There are several ways of combining two statements

The two more common are \[A\text{ and }B \qquad A\text{ or }B\]

The first one is sometimes called conjunction

The second one is called disjunction

Conjunction

As you can guess, \[A\text{ and }B\] is true when both \(A\) and \(B\) are True

Truth tables

This is an important tool

We can write all combinations of \(A\) and \(B\) and evaluate the value of \(A\text{ and }B\) in each case

We will make many truth tables

Exercise: How many rows are needed for any truth table?

Silly properties

It is easy to see that \[A\text{ and }B = B\text{ and }A\] (i.e. conjunction is commutative) \[(A\text{ and }B)\text{ and }C = A\text{ and }(B\text{ and }C)\] (i.e. conjunction is associative)

Associativity is good

Since conjunction is associative we can write \[A\text{ and }B\text{ and }C\] without parenthesis, since there is no ambiguity

This expression will be true when all the statements are True

Exercise: write the truth table so you can convince that conjunction is commutative and associative

Disjunction

As you can guess, \[A\text{ or }B\] is true when either \(A\) or \(B\) are True

Exercise: Write the truth table for disjunction

Two ways of saying “or”

Everyday language can be ambiguous. We may say

• You can eat or drink
• You can eat chicken or pasta

In the first case you can do both.
In the second one you can only choose one

The last one is called exclusive or (xor)

Exercise: Write the truth table for xor

More silly properties

It is easy to see that \[A\text{ or }B = B\text{ or }A\] (i.e. disjunction is commutative) \[(A\text{ or }B)\text{ or }C = A\text{ or }(B\text{ or }C)\] (i.e. disjunction is associative)

Associativity is still good

Since disjunction is associative we can write \[A\text{ or }B\text{ or }C\] without parenthesis, since there is no ambiguity

This complex expression will be true when any of the statements is True

Combine carefully

Notice that \[(A\text{ and }B)\text{ or }C\] is not the same as \[A\text{ and }(B\text{ or }C)\]

Thus, we need parenthesis when we combine “and” with “or”

Exercise: Write the truth table and verify that they are different

Distributivity

So, what is \(A\text{ and }(B\text{ or }C)\)?

This is a little like \(A⋅(B+C)\) which is equal to \(A⋅B + A⋅C\)
so we would expect a similar formula

Indeed, if we build the truth table, we can see that \[A\text{ and }(B\text{ or }C) = (A\text{ and }B)\text{ or }(A\text{ and }C)\]

Exercise: Write the truth table and verify that they are the same

There is more

We have seen that \[A\text{ and }(B\text{ or }C) = (A\text{ and }B)\text{ or }(A\text{ and }C)\] which is similar to the formulas with numbers

But these are not numbers, so there are different formulas

For instance we also have \[A\text{ or }(B\text{ and }C) = (A\text{ or }B)\text{ and }(A\text{ or }C)\]

Combining with “not”

We can write \[\text{not }(A\text{ and }B)\] and see that it is different from \[(\text{not }A)\text{ and }B\] Again, we must be careful and use parenthesis

Exercise: Write the truth table and verify that they are different

De Morgan’s Law

These two rules are super powerful

\[\begin{aligned} \text{not }(A\text{ and }B) &= (\text{not }A)\text{ or }(\text{not }B)\\ \text{not }(A\text{ or }B) &= (\text{not }A)\text{ and }(\text{not }B) \end{aligned}\]

In English

• The negation of a disjunction is the conjunction of the negations
• The negation of a conjunction is the disjunction of the negations