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Geometry/Chapter 3

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Geometry/Chapter 3 is about logical arguments.

Symbolic logic

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An if-then statement or conditional statement is a statement formed when one thing implies another, but not necessarily the other way around. For example, if the electrical power goes out then you will not be able to use your computer. (When given premises like this, always assume they are true — conveniently ignore the fact that your computer may be a laptop with a battery). However, if you are unable to use your computer, that does not automatically mean that the power went out. It only means that the power may have gone out.

But what if your computer is usable? Then we know that there is power—after all, if there wasn't, you wouldn't be able to use the computer. This is known as a contrapositive. A contrapositive is formed by turning an if-then statement around and negating both parts, and it is always true given the truth of the original statement.

In order to understand contrapositives on a mathematical level, you need to know about implication. Implication is a way to phrase if-then statements to indicate that one thing implies another. It is represented by an arrow, here typeset as "→". A statement using this arrow is known as a conditional statement, because the truth of the second value is conditional on the truth of the first. Not having electricity implies not being able to use your computer. Implication statements are only false when the first condition is true and the second condition is false. If both are true, it holds. If the first condition is false, the statement is considered vacuously true no matter what the second condition is.

You also may notice that both the conditions are negated. Negation is used to show that a condition is not true (also known as false), and is indicated by the "~" (tilde) symbol in front of the variable. Whether a given condition is true or false is known as its truth value. Note that a condition, represented by a variable, cannot be true and false at the same time. Anything that is always false is called a fallacy. Something that is always true is called a tautology. So not having electricity implying not being able to use your computer translates into symbolic logic as ~E → ~C. Symbolic logic is a system of logic using variables to represent conditions and symbols to represent the way the conditions are related. Now that we have put our conditional statement into symbolic logic, it is easy to see that the contrapositive is C → E.

You may also notice that there are other ways you can rearrange the variables and the tilde ("not" sign). One is to take the original expression and negate both sides, like you would an equation. Note that negating an already negated variable results in the removal of the not sign. This gives you E → C. This is known as the inverse; it inverts the truth values of the original statement. You also can switch both sides around, and not negate them like you would in a contrapositive. This gives ~C → ~E. This is known as the converse; it switches the values around. It's easy to see that these are contrapositives of each other, but are they the same as the original statement? No. This will be demonstrated by a truth table. A truth table is a table that accounts for all possible values (true or false) of all variables, and then gives results about whether a statement is true or false, given specific values for the variables.

First, let's review the nature of conditional statements, in table form. The truth tables we'll be using will use "F" to denote a false truth value and "T" to indicate a true truth value. We'll also use the generic conditional statement, p → q.

Logical Implication
p q p → q
F F T
F T T
T F F
T T T


Now let's look at the truth table for the original statement, inverse, converse, and contrapositive. For clarity, we'll include the negated forms of both variables.

Conditionals, inverses, converses, and contrapositives
p ~p q ~q p → q ~p → ~q q → p ~q → ~p
F T F T T T T T
F T T F T F F T
T F F T F T T F
T F T F T T T T

Since the original statement and the contrapositive have the same truth values in all the rows, they are proven to be equal. The same applies to the inverse and the converse, which are equal to each other, but not the other two. A truth table is one way to prove a concept, and it is the only way to prove something in symbolic logic (though you don't always have to write the whole thing out). Notice, however, that if we know p and q have the same truth value (which value doesn't matter), both the original statement/contrapositive and the inverse/converse are true. That is, p → q and q → p. This can be simplified into a biconditional. A biconditional is when both a conditional statement and its converse are true, and it can be written p ↔ q. If you know a biconditional is true, you can also use one or both of the conditional statements that go into it. A truth table is one way to prove a concept, and it is the only way to prove something in symbolic logic.

We have just completed our first proof. In mathematics, proof uses logic, rather than observation, to state definitively that something is always true. (You can also prove something impossible and it's still considered proof). Something that is proved false is said to be "disproved". This most often happens when someone finds a counterexample, which is where an exception is found and the formula does not hold, or work. The standards for proof in the mathematical community are very high, but anything that is proven is known to be true beyond a doubt. Anything that is proven is considered a theorem, and may be used in the proof of other theorems. Anything that is not proven is known as a conjecture.

Vocabulary

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  • Implication - When one condition is deducible based on another. Can be written p → q, and pronounced "If P then Q".
  • Truth Table - A way of visually representing a conditional for all values of the variables in that conditional.
  • Contrapositive - The conditional created when negating both sides of an implication. Can be written ~q → ~p, and said "If not Q then not P".
  • Fallacy - Incorrect reasoning in argumentation resulting in a statement that is not necessary true in all situations.
  • Contradiction - Something that is always false (e.g. caused by contradicting assumptions)
  • Tautology - Something that is always true.

Exercises

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1) If it is raining, then the dog is inside. Create a truth table for this statement.

2) If A, B, and C are the angles formed by a triangle, then A + B + C = 180° Create a truth table for this statement. (Advanced students will note that this isn't always true for non-euclidean geometries)


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