This is part 2 of a multi-part series aimed at increasing mathematical literacy and logical thinking by teaching the background for and then the process of proving mathematics theorems. But this is important for more than just mathematicians: the ability to think logically is the backbone of making strong arguments and is vital for every aspect of life.

If you missed [part 1, read it here.](https://steemit.com/mathematics/@daut44/an-introduction-to-mathematical-proofs-part-1-basic-logic-and-truth-tables)

## Summary of what was covered in part 1:
-**Propositions** are sentences that are either true or false.
-**Compound propositions** are finitely many propositions connected by logical symbols
-**Propositional forms** are expressions involving finitely many logical connectors and letters.
-a **paradox** is a sentence that leads to a self-contradictory conclusion.
-Use letters such as P, Q, R, S to shorthand propositions
-"P and Q" = P^Q, referred to as **conjunction**
-"P or Q" = P v Q, referred to as **disjunction**
-"not P" = ~P, referred to as **negation**
-truth tables with N propositions have 2^N lines
-When two compound propositional forms have identical truth tables they are **equivalent**
-**Tautologies** are always true regardless of truth assignments to individual propositions, i.e. P v ~P
-**Contradictions** are the negation of tautologies. i.e. ~(P v ~P)

## Conditionals
The most important kind of proposition in mathematics is a sentence of the form “If P, then Q”. This can be thought of as “P implies Q” or **“if P is true then Q is necessarily true”**. For example, “if X = 6, then 5 * X = 30” is a true sentence of this form because when X is 6, 5X is 30. But “if X = 6, then 5 * X = 20” is a false sentence of this form, because when X is 6, 5X is not 20. 

*Basically, a statement of the form “if P then Q” is true when the truth of P forces the truth of Q.*

### Definitions:
**conditional sentence:** Given propositions P and Q, "P => Q" is the proposition “if P then Q”. We use the **“=>”** symbol to denote “implies”.
**antecedent**: in the above example, the proposition P, or what is assumed.
**consequent**: in the above example, the proposition Q, or what is **deduced.**

Think of the conditional sentence in the logical form “if P is true then Q is necessarily true”.  P => Q is true whenever P (the antecedent) is false or Q (the consequent) is true. If P is false then the value of Q can be anything since our assumption is not met, but if P is true then Q *must* be true for P=>Q to be true. 

#### Truth Table for P => Q:
https://www.liquidpoker.net/staff/Daut/Math/p_implies_q.png

The truth table for P => Q is only false when P is true and Q is false. In every other case the “if P is true then Q is necessarily true” requirement is met. So logically, the conditional is equivalent to “not P or Q”: either P is false or Q is true”.

#### A check on the truth table of ~P v Q shows it is exactly equal to P => Q:
https://www.liquidpoker.net/staff/Daut/Math/p_or_not_q_same_as_p_implies.png


## Modus Ponens Deduction
If we assume that both P and P => Q are true, then we can **deduce** Q is true:
https://www.liquidpoker.net/staff/Daut/Math/modus_ponens.png

This is essentially a larger conditional in the form [P ^ (P => Q)] => Q, with [P ^ (P => Q)] as the antecedent, and Q as the consequent. Just as before, to verify this conditional is true, we only need to check that when the antecedent is true, the consequent is necessarily true. We ignore the 3 cases where the antecedent is false, and in the one case when it is true, Q was true as well. Thus, we have *deduced* that when P and P=>Q are both true that Q is true too.

This is our first rule of deduction and is referred to as **modus ponens**: this is latin for “the way that affirms by affirming”. It is incredibly obvious: we say “if P is true then Q is true”, and then we assume P is true, we know that Q is true. It’s the simplest deduction we can make, but is incredibly important for building up to more complex rules and deductive logic.

This will be the basis for our first proof technique later on: **the direct proof.**

## Converse and Contrapositive
For propositions P and Q, the **converse** of “P => Q” is “Q => P” and the **contrapositive** of “P => Q” is “~Q => ~P”. 

Example:
Let P = “X = 6”
Let Q = “X^2 = 36”
The *conditional*, or P => Q is “if X is 6 then X squared is 36”.
The *converse*, or Q => P is “X squared is 36 implies that X is 6”.
The *contrapositive*, or ~Q => ~P is “If X squared is not 36, then X is not 6.

Note that in this case the *converse* is false: if X^2 is 36, then X *could be* either positive or negative 6 since (-6)^2 = 36 as well as (6)^2=36. Thus, the converse is **false**, and it is not necessarily true if the conditional is true. 

But the contrapositive remains true: if X squared isn’t 36 then X cannot be 6. 

Let’s prove these two theorems using proof tables:
The propositional form P => Q is not equivalent to its converse, Q => P.
The propositional form P => Q is equivalent to its contrapositive, ~Q => ~P.

### Truth Table For Converse:
https://www.liquidpoker.net/staff/Daut/Math/converse.png

The truth tables for P=>Q and Q=>P are different in two spots, so they are not equivalent.

### Truth Table For Contrapositive:
https://www.liquidpoker.net/staff/Daut/Math/contrapositive.png

The truth tables for P=>Q and ~Q=> ~P are the same on every line, so they are equivalent. 

The equivalence of a conditional sentence and its contrapositive will be the basis for another one of the proof techniques we use a few lessons down the road, called **Proof by Contraposition.** Essentially, we can prove a statement of the form P=>Q if we prove that not Q implies not P since they are equivalent.

## Biconditional Sentences
A **biconditional** sentence P <=> Q is the combination of both P=>Q and Q=>P. The double arrow <=> is a shorter way of saying (P=>Q)^(Q=>P). In english, P <=> is the proposition **”P if and only if Q”**. The sentence is true exactly when P and Q have the *same truth values.*

### Truth Table for “if and only if”:
https://www.liquidpoker.net/staff/Daut/Math/iff.png
*Mathematicians abbreviate “if and only if” as “iff” for shorthand purposes*

Example:
“A rectangle is a square iff all four sides of the rectangle have equal length.”
A rectangle being a square implies that all four sides have equal length. But all four sides of a rectangle having equal length implies that the rectangle is a square. Thus P=>Q and Q=>P so P<=>Q, or P iff Q.

## Equivalences
For propositions P, Q, and R, the following are equivalent:
1. P => Q is equivalent to (~P) v Q
2. P <=> Q is equivalent to (P=>Q) ^ (Q=>P)
3. ~(P^Q) is equivalent to (~P) v (~Q)
4. ~(PvQ) is equivalent to (~P) ^ (~Q)
5. P^(QvR) is equivalent to (P^Q) v (P^R)
6. Pv(Q^R) is equivalent to (PvQ) ^ (PvR)

We already proved #1 and #2 using truth tables before.

### #3 and #4 are **De Morgan’s Laws.** Let’s prove #3 using a truth table (#4 will have a similar proof):
https://www.liquidpoker.net/staff/Daut/Math/demorgan.png

De Morgan’s Laws **distribute** negation over *conjunction* in #3 and *disjunction* in #4.

### #5 and #6 are **Distributive Laws.** Let’s prove #5 using a truth table (#6 will have a similar proof):
https://www.liquidpoker.net/staff/Daut/Math/distributive_law.png

Armed with these equivalences, we can now prove other equivalences *without the use of truth tables.* For instance, suppose we want take a look at ~(P=>Q) and want to find out what it is equivalent to:
~(P => Q) is equivalent to ~(~P v Q) via #1.
~(~P v Q) is equivalent to P^~Q via #4.
Thus by the **transitive property**, ~(P => Q) is equivalent to P ^ (~Q).

The equivalences you can prove will grow steadily more complex, but using a set of basic rules we can now perform proofs without truth tables.

## Translating English Phrases Using Logical Symbols
Shown below are phrases in English that are usually translated using connectives => and <=>, with examples to help make sense of them.

### Use P=>Q to translate:
**"if P, then Q”**
	Example: if x>10 then x>7.

**"P implies Q”**
	Example: x>10 implies x>7.

**"P is sufficient for Q”**
	Example: x>10 is sufficent for x>7

**"P only if Q”**
	Example: x>10 only if x>7

**"Q, if P”**
	Example: x>7 if x>10

**"Q whenever P”**
	Example: x>7 whenever x>10

**”Q is necessary for P”**
	Example: x>7 is necessary for x>10

### Use P <=> Q to translate:
**"P if and only if Q”**
	Example: |x|=3 if and only if x^2=9

**”P is equivalent to Q”**
	Example: |x|=3 is equivalent to x^2=9


## Summary:
-**conditionals** have the form “if P then Q”
-**antecedent** refers to the proposition P, or what is assumed.
-**consequent** refers to the proposition Q, or what is deduced.
-**Modus Ponens** is the first deductive logic rule, which states that if we know P and P=>Q then we can deduce Q. This sets up our first future proof technique, **The Direct Proof**.
-the **converse** of P=>Q is Q=>P, and it is **not equivalent** to the conditional.
-the **contrapositive** of P=>Q is ~Q => ~P, and it is **equivalent** to the conditional P=>Q. This sets up our second future proof technique, **The Proof by Contraposition**.
-**biconditional** sentences have the form “P if and only if Q”
-**De Morgan’s Laws** are two famous equivalences:
~(P^Q) is equivalent to (~P) v (~Q)
~(PvQ) is equivalent to (~P) ^ (~Q)
-**The distributive Laws are two famous equivalences:
P^(QvR) is equivalent to (P^Q) v (P^R)
Pv(Q^R) is equivalent to (PvQ) ^ (PvR)

## Other great Mathematics Articles I’ve read:
[An introduction to group theory](https://steemit.com/mathematics/@complexring/cryptography-101-an-interactive-class-an-introduction-to-group-theory--week-1) by @complexring
[Part 2 of introduction to group theory](https://steemit.com/mathematics/@complexring/cryptography-101-an-interactive-class-more-introduction-to-groups-week-2) by @complexring
[Pascal’s Triangle - The “Swiss Army Knife” of Mathematical Tools](https://steemit.com/mathematics/@team-leibniz/pascal-s-triangle-the-swiss-army-knife-of-mathematical-tools) by @team-leibniz
[L’Hopital’s Rule - A 17th Century Story of Paying for Content Curation](https://steemit.com/mathematics/@team-leibniz/l-hopital-s-rule-a-17th-century-story-of-paying-for-content-curation) by @team-leibniz

I welcome any other math threads or people I should follow in the comment section.

___________________________________________________________________________________________________
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