In essence, mathematics is the development of new knowledge, by logical deduction, from old knowledge. The new knowledge is called the conclusion (or the consequence) and the old knowledge is called the premises (or the assumptions or the antecedent)\(^1\) . Our next definition introduces the vehicle that we will use to conclude new knowledge from old.

\(^1\) These definitions will be given shortly.

DEFINITION  Theorem
A theorem is a proposition that has been, or is to be, proved on the basis of explicit assumptions.
A theorem is a proposition that has been, or is to be, proved on the basis of explicit assumptions.
Like many definitions, this seems a little difficult to understand without some clearer explanation. The definition is essentially saying that theorems are claims that can be proven using previous information someone gave us. Let's take a look at a simple theorem which is often cited in textbooks on logic.
THEOREM
If it is raining, then it must be cloudy. This is a theorem because someone, namely me, is proposing that for it to rain, we must have clouds. Can this be proven? Since I am not an expert in meteorology, I cannot back my claim other than to use a common sense approach. Nonetheless, as long as my argument has no flaws, this theorem would hold true.

Historical Note 
In mathematics, the theorem is king! It is the basis of correctness and there is no statement stronger than the theorem (as long as it has been proved, of course). I often tell my students that you must not believe anything unless it has been proven to you. While this is admittedly extreme, it is a wise suggestion when dealing with mathematics.
It turns out that there is structure to a theorem. They have two parts, both of which we have already mentioned. These are the assumption and the conclusion.
It turns out that there is structure to a theorem. They have two parts, both of which we have already mentioned. These are the assumption and the conclusion.
DEFINITION  Assumption and conclusion of a theorem
The assumption (also known as the antecedent) of a theorem is a formal set of conditions that we accept as truth for the statement. The conclusion (also known as the consequence) of a theorem is the result that can be derived from the assumption according to a logical string of arguments.
The assumption (also known as the antecedent) of a theorem is a formal set of conditions that we accept as truth for the statement. The conclusion (also known as the consequence) of a theorem is the result that can be derived from the assumption according to a logical string of arguments.
If you think about this definition, it is essentially stating that the first part of a theorem is some type of assumption. This assumption is often considered the "if" statement. The second part of a theorem is what we think should happen as long as our assumption holds. This conclusion is often considered the "then" statement. In fact, it is often easier to rewrite a theorem in the form "If…, then…" in order to determine the assumption and the conclusion. When written in this form, the statement following the word "if" is the assumption/antecedent and the statement following the word "then" is the conclusion/consequence.
EXAMPLE 1  Identifying assumptions and conclusions
In the following theorem, what is the assumption and what is the conclusion?
In the following theorem, what is the assumption and what is the conclusion?
"If it is raining, then it must be cloudy."


SOLUTION 1
The words following "if" are "it is raining." Therefore, the assumption is that it is raining. The words following "then" are "it must be cloudy." Hence, the conclusion is that it must be cloudy.
The words following "if" are "it is raining." Therefore, the assumption is that it is raining. The words following "then" are "it must be cloudy." Hence, the conclusion is that it must be cloudy.
Just a single example is not enough to do justice to the idea of assumptions and conclusions. It would also be nice to see some examples involving the other terms (antecedent and consequence). Let’s try another one.
EXAMPLE 2  Identifying assumptions and conclusions
In the following theorem, what is the antecedent and what is the conclusion?
In the following theorem, what is the antecedent and what is the conclusion?
"If I drink 14 gallons of water within 10 minutes, I will die."
SOLUTION 2
This is nearly written in “If…, then” form. Rewriting it as such, we get
This is nearly written in “If…, then” form. Rewriting it as such, we get
"If I drink 14 gallons of water within 10 minutes, then I will die."
The antecedent is that I drink 14 gallons of water in 10 minutes. The conclusion, based on that assumption, is that I will die.
The next example illustrates that some critical thinking will be necessary to reword sentences or claims into proper form so we can see the assumptions and conclusions.
EXAMPLE 3  Identifying hidden assumptions and conclusions
The following is a statement a mother made to her son. "Eating ice cream causes polio."
What is the assumption and what is the consequence?
SOLUTION 3 In this statement (theorem), we are completely missing the words "if" and "then." However, we can always reword it to contain these key words. "If you eat ice cream, then you will get polio."
Now we can easily see that the assumption is that the son is eating ice cream and the consequence is that doing so will cause polio.

Historical Note
A longterm study released in the late 1940's claimed this very statement to be true. While there are still those who cling to this belief, it is widely accepted that the interpretation of the data was flawed. First, polio is a communicable viral infection. Second, the high values of polio cases and ice cream consumption were from data collected during the summer. Of course, kids play in groups more during the warmer weather months and it is fairly obvious that warmer seasons will increase ice cream consumption. Finally, the low values of polio cases and ice cream consumption were from data collected during the winter. Again, this is pretty obvious – ice cream sales drop during colder months and children tend to play less in groups when it is cold outside. 
In other fields, terms like conjecture or hypothesis are used instead of theorem; however, each of these implies that the statement has not yet been proved and therefore the conclusion should be suspect. In fact, only in mathematics do we have the pleasure of the theorem.
Since I am not speaking "mathspeak" right now, the examples we have seen so far cannot officially be called theorems – they are technically conjectures or hypotheses. This is because someone could find a reason why the conjecture is false (remember, theorems have already been proved). Despite this, to keep our language less confusing, I will continue to use the word theorem to describe these and all future examples.
Since I am not speaking "mathspeak" right now, the examples we have seen so far cannot officially be called theorems – they are technically conjectures or hypotheses. This is because someone could find a reason why the conjecture is false (remember, theorems have already been proved). Despite this, to keep our language less confusing, I will continue to use the word theorem to describe these and all future examples.
EXAMPLE 4  "Breaking Bad" theorems
In each of the following theorems, state the antecedent and the consequence. Then state a possible reason why the theorem could be false.
SOLUTION 4
In each of the following theorems, state the antecedent and the consequence. Then state a possible reason why the theorem could be false.
 If Gage eats the ice cream cake, then he is going to gain weight.
 You will get ticketed if you speed down this street.
 Calculus II is a course that all engineering students at Cosumnes River College must take.
SOLUTION 4
 The antecedent is that Gage is going to eat ice cream cake. The consequence is that he is going to gain weight. Is this theorem necessarily true? If Gage plans on working out a lot more than he normally does after he eats the ice cream cake, then it could be a false statement.
 This is an example where rewriting as an "if…, then…" statement will help out immensely. The statement is saying, "If you speed down this street, then you will get ticketed." Thus, the antecedent is that you are speeding down the street and the consequence is that you will be ticketed. Can you think of a way that this does not need to be true?
 The way this is written can throw some students off, but this is definitely a theorem. The antecedent is that we have an engineering student at Cosumnes River College and the consequence is that she must take Calculus II ("If you are an engineering student at Cosumnes River College, then you must take Calculus II"). There is an easy way around this, though. Suppose the student reads a calculus textbook over the summer and tests out of Calculus II. Then she successfully avoided taking the course and so the theorem has a flaw.
As you can see from these examples, "theorems" outside of mathematics often have logical loopholes. Fortunately, true theorems (those within the field of mathematics) are generally flawless.
You should now have a basic grasp of what a theorem is; however, understanding the significance of theorems in mathematics is incredibly important. So, just how big of a deal are theorems in mathematics? Well, you might say that without theorems there is no thought. Without theorems, the world is just a bunch of statements with no conclusions. The idea of a theorem allows us to say,
You should now have a basic grasp of what a theorem is; however, understanding the significance of theorems in mathematics is incredibly important. So, just how big of a deal are theorems in mathematics? Well, you might say that without theorems there is no thought. Without theorems, the world is just a bunch of statements with no conclusions. The idea of a theorem allows us to say,
"If it is raining, then it must be cloudy."


This statement makes sense and almost serves as advice or warning. Without the theorem structure, either the conclusion or the assumption is gone and we would only say,
"It is raining." or "It is cloudy."
These are just exclamations without any conclusion. It sounds more like we are stating a fact than suggesting what would happen "if" a condition was met.
As another example, consider the "theorem" mentioned in EXAMPLE 4  "Breaking Bad" theorems.
As another example, consider the "theorem" mentioned in EXAMPLE 4  "Breaking Bad" theorems.
"You will get ticketed if you speed down this street."
As we stated, the assumption here is that you are speeding down the street and the conclusion is that you will get ticketed. Removing the assumption leads to the odd statement,
"You will get ticketed."
Removing the conclusion leads to an even more nonsensical statement,
"You speed down this street."
In either case, these are just statements without any assumptions or conclusions. They offer no advice, no warning, and no way out. With the assumption removed, you are getting ticketed – period! With the conclusion removed, you are definitely speeding down the street.
The next example illustrates a method to help determine whether or not a statement is a theorem. We essentially are trying our best to put it in the "if… then…" form. While more complicated theorems are difficult to place into this form, the examples we see in this textbook are pretty nice to work with and can be manipulated into this form somewhat easily.
The next example illustrates a method to help determine whether or not a statement is a theorem. We essentially are trying our best to put it in the "if… then…" form. While more complicated theorems are difficult to place into this form, the examples we see in this textbook are pretty nice to work with and can be manipulated into this form somewhat easily.
EXAMPLE 5  Identifying theorems
Which of the following has the structure to be a theorem?


SOLUTION 5
 This cannot be written in the "if…, then…" form so it definitely is not a theorem. It would be a theorem if it were rewritten as "If Akash has the winning lottery ticket, then he has at least a million dollars."
 This is a theorem because we can rewrite this as, "If Lisa goes to the store, then she spends too much money."
 This is a far cry from a theorem. How the heck can you fit an "if… then…" statement here?

