Check over your proof using the
checklist points, and check for readability.
## Some Types of Proofs

### Proof of A and B

To prove A and B, we must prove that both of them always hold.

### Proof of A or B

To prove A or B, we must prove that at least one of them holds
in any given situation (This does not mean that A always
holds - it could be false some of the time and true the rest
of the time, as long as B is true whenever A is false.
Note that if we have already
proven A then A OR B is true by default, since one of A or B
(A in this case) always holds in any situation.)

### Proof of A ---> B

Assume that A is true and prove that B is true.

### Proof of A <---> B (A If and Only If B)

Prove A --->B
and then also prove B--->A (ie A<---B)

### Negations

For all, every, *turns into* there is or there exits

There is or there exists *turns into* for all, every

A AND B *turns into* ~A OR ~B

A OR B *turns into* ~A AND ~B

A--->B *turns into* A AND ~B

A <--->B (ie A--->B AND B--->A) *turns into* (A AND ~B) OR (B AND ~A).
### Proof By Contradiction

To prove something via contradiction, we assume the negation of
the statement and eventually arrive at a contradiction.

### Proof of a Statement By Examining All Possible Cases

Sometimes one must examine lots of cases in order to prove
a statement - for example, one might have different proofs
for a statement about numbers which depend on whether a certain
number is zero
or not.
## Samples of Proofs and *Comments in Italics*

### Problem 1

Prove that if a minesweeper
square S is a 1, and an adjacent
square to S is a bomb, then every other square adjacent to
S must be a number.
#### Proof of Problem 1

Assume that S is a minesweeper square with a 1 in it, and
that some adjacent square to S, call it B, is a bomb. We will show
that every other square adjacent to S must be a number.
By definition of the type of a minesweeper square, this
is equivalent to showing that every other square adjacent to
S cannot be a bomb, since every square is either a bomb or
a number. *Notice that I have reworded
the desired conclusion in terms of the definitions.*
Assume for contradiction that some other square adjacent
to S, call this square P, is a bomb.
*Notice that I have assumed the negation of the desired
result for contradiction - every other adjacent square cannot be
a bomb ***turns into** there is an adjacent square which is a bomb.

Now P and B are both distinct bombs adjacent to S, by assumption, and
S is not a bomb, since it has a 1 in it by assumption,
and so by minesweeper rules,
we know that S is a number which must be at least 2. Yet, S
has a 1 in it, and so we have arrived at a contradiction
to the fact that some other square adjacent to S is a bomb, as desired.

Therefore,
if a minesweeper square S is 1
and an adjacent square is a bomb,
then every other square adjacent to S is a number.

### Problem 2

Prove that if we have a 2x2 minesweeper game
where A1=1 and A2=* then B1 is 1 AND B2 is 1.
#### Proof of Problem 2

Assume that we have a 2x2 minesweeper game with A1=1 and A2=*.
We must show that B1=1 and B2=1, ie we must show that
B1 and B2 are not bombs and that they each have exactly
1 bomb near them.
*Notice that I have reworded
what it means for a square to be 1 in terms of the definitions.*
We will first show that B1 and B2 are not bombs.
Since A1 is 1 and A2 is a bomb adjacent to it, we can apply
problem 1 to see that any other adjacent square to A1 cannot be a
bomb.
*Notice that I used proposition 1.
I had to check that the assumptions of prop 1 were satisfied.
They were, so prop 1 gave me the conclusion that any other adjacent
square was not a bomb. *

Since B1 and B2 are both adjacent to A1, then we know that
they cannot be bombs, as desired.
*I then used the fact that B1
and B2 are both adjacent to A1 to get the desired conclusion.*

We will now show that B1 and B2 are both 1.
Notice that B1 has A1=1, A2=*, and B2=number adjacent to it,
and B1 is a number since it is not a bomb, and so B1=1 since it
it is a number and it has exactly one bomb next
to it.
*Notice that I wrote out all the details to
prove that B1 is 1. It would not have been ok to
say that it is obvious that B1 is a 1. *

Similarly, B2=1, since B2 is not a bomb
and A1=1, A2=*, and B1=1
are the adjacent squares to it.
*Since the proof that B2=1 is basically the same,
I do not have to write it all out again, but I do need to include
enough detail so that the similarity is clear*

Therefore,
if we have a 2x2 minesweeper game with A1=1 and A2=*,
then B1=1=B2, as desired.