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# Propositional Logic

## What are Propositions?

Propositions are anything that can be true or false. This could include:
• Statements like "Birds can fly".
• Well defined equations with no free variables like
$1 + 1 = 3$
.
Propositions are not:
• Variables like
$x$
or
$5$
.
• Equations with free variables like
$P(x) = y$
.
• Statements that aren't clearly true or false, like "I like trains."

### Connectives

Simple propositions can be joined together to make complex statements. There are three basic ways to connect propositions together:
• Conjunction is the and operation: for
$P \land Q$
to be true,
$P$
and
$Q$
must both be true.
• Disjunction is the or operation: for
$P \lor Q$
to be true, either
$P$
or
$Q$
must be true.
• Negation is the not operation: if
$P$
is true, then
$\lnot P$
is false.
• The law of the excluded middle states that
$P$
and
$\lnot P$
cannot both be true.
One example where we can see these components in action is in De Morgan's Laws, which state how negation can be distributed across conjunction or disjunction:
$\lnot(P \lor Q) \iff (\lnot P \land \lnot Q)$
"If neither P nor Q are true, then P and Q must both be false."
$\lnot(\forall x)(P(x)) \iff (\exists x)(\lnot P(x))$
"If P(x) isn't true for every x, then there exists an x where P(x) is false."

Another example of distribution is this congruence, which works for any combination of and's and or's.
$(P \lor Q) \land R \equiv (P \land R) \lor (Q \land R)$

### Implication

One proposition can imply another, which looks like this:
$P \implies Q$
Roughly, implication in plain English can be stated in the form if P, then Q. However, there are a lot of nuances to what this really means!

#### Properties of Implication

• Reversible: Q is true if P is true. However, be careful- this doesn't necessary mean that Q implies P!
• P is sufficient for Q: Proving P allows us to say that Q is also true.
• Q is necessary for P: For P to be true, it is necessary that Q is true. (If Q is false, then P is also false.)
• Contrapositive Equivalence: If P implies Q, then
$\lnot Q \implies \lnot P$
.
• Note that this is different from the converse, which is
$Q \implies P$
. This statement is not logically equivalent!

#### Truth Table

 P Q P $\implies$Q P $\iff$Q T T T T T F F F F T T F F F T T
Note that the truth table for
$P \implies Q$
is equivalent to the one for
$\lnot P \lor Q$
! That means this formula is logically the same as
$P \implies Q$
.
(If two propositions have the same truth table, then they are logically equivalent. However, it's still possible for a proposition to imply another even if their truth tables are different!)

### Quantifiers

Sometimes, we need to define a specific type of variable to work with in a propositional clause. For instance, take the proposition, "There exists a natural number that is equal to the square of itself." We could write this as:
$(\exists x \in \mathbb{N})(x=x^2)$
You could think about the parentheses almost like defining a scope of variables, like what might happen in programming! Here, the first clause is defining an arbitrary variable
$x$
to be any natural number.

## Exercises

Q1
Is the expression
$\forall x \exists y (Q(x,y) \implies P(x))$
equivalent to the expression
$\forall x ((\exists y \ Q(x,y)) \implies P(x))$
? (Source: Discussion 0 2a)
No, they are not equivalent. We can see this more clearly by converting the implication
$Q \implies P$
to
$\lnot Q \lor P$
as was demonstrated in the Truth Table section above. On the left side, this conversion is straightforward, yielding
$\forall x \exists y (\lnot Q(x,y) \lor P(x))$
.
On the right side, we'll need to invoke De Morgan's Laws to convert the 'exists' into a 'for all' since it was negated. This yields
$\forall x (\forall y\lnot(Q(x,y)) \lor P(x))$
which is not the same thing!
Q2
An integer
$a$
is said to divide another integer
$b$
if
$a$
is a multiple of
$b$
. Write this idea out using propositional logic (a divides b can be written as
$a \mid b$
).
Note: This idea is going to be important for a lot of future sections!
$a \mid b \iff (\exists q \in \mathbb{Z})(a = qb)$
In plain English: "There exists an integer
$q$
such that when we multiply
$q$
with
$b$
, we get
$a$
."