To help you write strong proofs, we’re going to talk about logical reasoning, the foundation on which all valid proofs are built.

One key element of proofs is the premises which are used in them.

Premises are statements which are already assumed to be true, and from these true statements, new conclusions may be derived.

For example, we already know that the interior angles of a triangle must add up to 180 degrees.

This fact may be regarded as a premise.

Proofs as a whole may be referred to as arguments.

An argument is a collection of premises followed by a conclusion.

If the premises used in an argument are all true, and if the conclusion follows logically from the premises,

then the argument is considered valid and the proof is complete.

The individual sentences which compose a proof are called statements and may be either simple statements or compound statements.

Statements must have a truth value, meaning they must be able to be proven true or false.

“The sky is blue,” is a statement because it can be proven true.

“There are a lot of clouds in the sky,” is not considered a statement because “a lot” is not specifically measurable.

I may think there are a lot of clouds in the sky but you may disagree.

So, the statement cannot be proven true or false.

Opinions are never considered statements in math.

Simple statements are straightforward and are intended to convey one thought.

Compound statements, on the other hand, include one or more simple statements along with a logical operator.

The five logical operators are negation, conjunction, disjunction, conditional, and biconditional.

If that all sounded like gibberish to you, there’s no need to worry.

Let’s clear any confusion by going over what each of these mean.

The negation operator is denoted with a tilde, or squiggle, that precedes the simple statement, and its action is to undo, or give the opposite, of that statement.

For example, if we wanted to negate the statement, “There are no trucks in the parking lot,” this would be like saying,

“It is not true that there are no trucks in the parking lot.”

In other words, we could say, “There is at least one truck in the parking lot.”

To negate a statement is to claim that it is untrue.

The second type of logical operator is conjunction, which essentially connects two simple statements with the word “and.”

The symbol for conjunction is an upside-down “V”.

So, if we wanted to say, “There are no trucks in the parking lot and there are no cars in the parking lot,” we could write this.

This statement would be true only if the parking lot contained zero cars and zero trucks.

Notice that the conjunction symbol is similar to the symbol we use for the intersections of sets.

In fact, a conjunction may be thought of as a type of intersection, because we use the word “and” to show that two statements (like sets) are fulfilled simultaneously.

The third logical operator is disjunction, which is denoted with a V-shaped symbol, and means “or.”

If you guessed that disjunction is quite like the union of sets, you would be quite right!

Disjunctions are satisfied when at least one of the simple statements given is true.

For example, if we say, “There are no trucks in the parking lot or there are no cars in the parking lot,” then we are correct as long as one or both of those are true.

We are only wrong if both simple statements are false.

The fourth logical operator is the conditional, which we can think of as an “if, then” statement.

For example, the statement, “If it is a holiday, then there is no school”