For each of a handful of logical connectives, we summarized how to prove them, how to use them in a proof, and how to negate them.

We have seen many of these things in passing, either in lecture or in the homework, here we bring them together in one place.

A **proposition** is a statement that is either true or false. For example, "5^{2} = 25", "the pope is catholic", and "John can fool Mike" are all propositions.

A **predicate** is a statement whose truth depends on a variable. For example, "*x*^{2} = 25", "*x* is catholic", and "*x* can fool Mike" are all predicates.

You can think of a predicate as a function that returns a proposition.

Note that if *P* is a predicate, then ∀ *x*, *P*(*x*) and ∃ *x*, *P*(*x*) are propositions.

In the following, *P* and *Q* may denote either propositions or predicates.

logical statement | English interpretation | How to prove it | How to use it | Logical negation |
---|---|---|---|---|

P ∧ Q |
P and Q |
prove P and then prove Q. |
if you know P ∧ Q, you can conclude P. You can also conclude Q. |
(¬P) ∨ (¬Q) |

P ∨ Q |
P or Q (or both) |
either give a proof of P or give a proof of Q (your choice). |
You can do case analysis. If you know P ∨ Q and are trying to prove R, you can give one proof of R assuming P is true, and another proof of R assuming Q is true. This is enough to conclude that R is always true. |
(¬P) ∧ (¬Q) |

P⇒Q |
"if P then Q" or "P implies Q". Be careful because $P \implies Q$ does not say that there is any causal relationship between P and Q, just that if P happens to hold, then Q holds as well. Could be coincidence. |
Assume P, and then prove Q. |
If you know $P \implies Q$ and you know P you can conclude Q. |
P ∧ (lnotQ) |

∀ x ∈ A, P(x) |
for all x in A, P(x) is true. |
Choose an arbitrary x (see below). Using only the fact that x ∈ A, prove P(x). |
If you know ∀ x ∈ A, P(x) and you know some element y ∈ A, then you can conclude P(y). |
∃ x, ¬P(x) |

∃ x ∈ A, P(x) |
there is some x in A satisfying P. |
Write down a specific x. Check that x ∈ A, and prove P(x). |
If you know that ∃ x ∈ A, P(x), you can use x in the remainder of your proof, as well as the facts x ∈ A and P(x). |
∀ x, ¬P(x) |

true |
"true" | No effort required. | Not much you can do with it. | false |

false |
we have arrived at a contradiction | Good luck proving that false is true! | But if you come to a falsehood in your proof, you must have made a bad assumption, so you can conclude anything you want (proof by contradiction) | true |

Note that proving things is a recursive process: to prove a complicated statement involving *P* and *Q*, you may have to prove *P* or prove *Q*.

For example, to prove ∀ *ε* > 0, ∃ *n* ∈ N, ∀ *i* > *n*, 1 / 2^{i} < *ε*, You would first choose an arbitrary *ε*. Then, you prove ∃ *n* ∈ N, ∀ *i* > *n*, 1 / 2^{n} < *ε*.

To do that, you give a specific *n*. In this case, I know that there exists some *n* > *l**o**g*_{2}(1 / *ε*), so I will choose that *n* (here I am using an existential statement). Now I must prove that ∀ *i* > *n*, 1 / 2^{n} < *ε*.

To do that, I choose an arbitrary *i* and assume only that *i* > *n*. I must show 1 / 2^{n} < *ε*. But I know that *n* > *l**o**g*_{2}(1 / *ε*) and that *i* > *n*, so *i* > *l**o**g*_{2}(1 / *ε*). Then 2^{i} > 1 / *ε*, so 1 / 2^{i} < *ε*.

When you "choose an arbitrary *x*", you cannot assume anything about *x*. For example, it is wrong to say "choose an arbitrary *x*, say 7", because you are assuming many things about *x*: it is odd, it is prime, it is greater than 3, .... This means that your proof will not work if I want to apply it to a different *x*, so you haven't proven ∀ *x*, *P*(*x*), you've just proven *P*(7) (or ∃ *x*, *P*(*x*))

Similarly, when using an existential statement, you can't assume anything about the element that exists, other than that it has the given property.

Don't write backwards proofs. A backwards proof is one that starts with what you are trying to show, and ends with something you know. Backwards proofs make it easy to prove nonsense:

**Claim:** 5 = 3.

**Proof:**

5 = 3

3 = 5

5 + 3 = 3 + 5

8 = 8✓

It is often helpful to write down what you're trying to show, but if you do, you **must** clearly indicate this, for example by writing "we want to show that".