March 20, 2026
Kuina-chan
In Lesson 1 of “Kuina-chan Mathematics”, we will explain the rules and conventions of mathematics!
1.Axioms, Theorems, and Proofs
In mathematics, broadly speaking, we start from some premises that are assumed to be correct, and logically derive things that can be said to be correct. These pre-determined correct premises are called “axioms”.
In addition to axioms, some rules are defined, and in mathematics, we use axioms and these rules to derive correct things one after another.
The newly derived correct things, together with the axioms, are called “theorems”, and the process of deriving a theorem is called a “proof”.
From another perspective, solving a mathematical problem is the task of finding a proof of how the answer to the problem becomes a theorem, using the theorems derived so far.
2.Propositions and Logical Formulas
Now, objects that can be judged as to whether they are theorems or not, such as “it is 
” and “it is 
”, are called “propositions”.
” and “it is ”, are called “propositions”.There are several ways to handle propositions, but here, for simplicity, we will use “True” and “False” of logical formulas to express that “a proposition being a theorem is ‘True’, and not being a theorem is ‘False’”. For example, if the proposition “it is ” is a theorem, then “it is ” is “True”. If the proposition “it is ” does not become a theorem, then “it is ” is “False”.
” is a theorem, then “it is ” is “True”. If the proposition “it is ” does not become a theorem, then “it is ” is “False”.
Supplement
Formulas that handle True and False in this way are called “logical formulas”. This time, we decided to use the truth value of logical formulas to express whether a proposition is a theorem, but there are other ways to express whether a proposition is a theorem. For example, one idea is to consider a proposition that is always True, called a “tautology”, as a theorem.
At this time, we will represent propositions with letters such as “
” and “
”. Then, we consider creating new propositions by combining them, such as “if then ” and “ and ”.
” and “”. Then, we consider creating new propositions by combining them, such as “if then ” and “ and ”.For example, if is the proposition “it is ” and is the proposition “it is ”, by saying “ or ”, we can create the proposition “it is , or, it is ”.
is the proposition “it is ” and is the proposition “it is ”, by saying “ or ”, we can create the proposition “it is , or, it is ”.
Usually, “or” is represented by the “
” symbol, and “and” is represented by the “
” symbol, written as “” and “”. That is, the proposition “it is or it is ” can be written as “
”.
” symbol, and “and” is represented by the “” symbol, written as “” and “”. That is, the proposition “it is or it is ” can be written as “”.By the way, “ or ” means that it is True if either or is True. For example, if the proposition “it is or it is ” is True, it means that either “” or “” is True. In other words, the result of “” is as shown in the following table.
or ” means that it is True if either or is True. For example, if the proposition “it is or it is ” is True, it means that either “” or “” is True. In other words, the result of “” is as shown in the following table. |
|
|
|---|---|---|
| False | False | False |
| False | True | True |
| True | False | True |
| True | True | True |
On the other hand, “ and ” means that it is True if both and are True. In other words, the result of “” is as shown in the following table.
and ” means that it is True if both and are True. In other words, the result of “” is as shown in the following table. |
|
|
|---|---|---|
| False | False | False |
| False | True | False |
| True | False | False |
| True | True | True |
For example, suppose “” is True, that is, a theorem, and “” is False, that is, not a theorem. At this time, “” becomes “True and False”, which is False, meaning it is not a theorem.
” is True, that is, a theorem, and “” is False, that is, not a theorem. At this time, “” becomes “True and False”, which is False, meaning it is not a theorem.Supplement
To be precise, we have decided here that if a proposition created using “or” or “and” in a logical formula is True, it is a theorem. From now on, we will similarly decide that what becomes True in a logical formula is a theorem.
3.Properties of Logical Formulas
From here, we will explain various properties of logical formulas that are necessary when proving theorems.
3.1Negation, Law of Excluded Middle, and Contradiction
When expressing a negative proposition “it is not ” against the proposition “it is ”, we use the “
” symbol. For a proposition , “not ” is written as “”, and the result at that time is as shown in the following table.
” against the proposition “it is ”, we use the “” symbol. For a proposition , “not ” is written as “”, and the result at that time is as shown in the following table. |
|
|---|---|
| False | True |
| True | False |
From this table, we can see that for any proposition , either “” or “” is True, that is, it becomes a theorem. In other words, there is no proposition where neither “” nor “” is a theorem. This law that “there is no proposition where neither nor becomes a theorem” is called the “law of excluded middle”.
, either “” or “” is True, that is, it becomes a theorem. In other words, there is no proposition where neither “” nor “” is a theorem. This law that “there is no proposition where neither nor becomes a theorem” is called the “law of excluded middle”.On the other hand, the fact that “both and are theorems” is called a “contradiction”. From this table, we can also see that there is no proposition that causes a contradiction.
and are theorems” is called a “contradiction”. From this table, we can also see that there is no proposition that causes a contradiction.By combining the law of excluded middle and contradiction, we can also prove its negation by intentionally causing a contradiction, such as “if we assume is a theorem, it contradicts, therefore is a theorem”.
is a theorem, it contradicts, therefore is a theorem”.3.2Logical Implication
As another symbol for logical formulas, there is “
” which means “if then ”. This is a proposition that “when holds, holds”.
” which means “if then ”. This is a proposition that “when holds, holds”.
The fact that the proposition “” is a theorem means that whenever “ is True”, “ is also True”.
” is a theorem means that whenever “ is True”, “ is also True”.At this time, when “ is False”, it does not matter what is. In other words, when “ is False”, no matter what is, the fact that “” is a theorem is not overturned, so at this time “” can be said to be True.
is False”, it does not matter what is. In other words, when “ is False”, no matter what is, the fact that “” is a theorem is not overturned, so at this time “” can be said to be True.That is, if of “” is False, “” is True whether is True or False. It is as shown in the following table.
of “” is False, “” is True whether is True or False. It is as shown in the following table. |
|
|
|---|---|---|
| False | False | True |
| False | True | True |
| True | False | False |
| True | True | True |
For example, when there is a theorem “if 
, then is an odd number”, it says nothing about the case where is not , so if is not , whether is an even number or an odd number, this theorem will not be overturned. Therefore, we can understand that when “if False, then...”, this proposition should always be True.
, then is an odd number”, it says nothing about the case where is not , so if is not , whether is an even number or an odd number, this theorem will not be overturned. Therefore, we can understand that when “if False, then...”, this proposition should always be True.3.3Equivalent Propositions
Now, when the truth values of propositions 
always match, and are said to be “equivalent”, and we write “”.
always match, and are said to be “equivalent”, and we write “”.If becomes a theorem when is a theorem, and becomes a theorem when is a theorem, it can be said that the truth values of and match, so and are equivalent. In other words, written as a logical formula, when “”, and are equivalent. For this reason, “” is sometimes written with the symbol “
”.
becomes a theorem when is a theorem, and becomes a theorem when is a theorem, it can be said that the truth values of and match, so and are equivalent. In other words, written as a logical formula, when “”, and are equivalent. For this reason, “” is sometimes written with the symbol “”.If and are equivalent, proving either one means proving the other as well. The result of “” is as shown in the following table.
and are equivalent, proving either one means proving the other as well. The result of “” is as shown in the following table. |
|
|
|---|---|---|
| False | False | True |
| False | True | False |
| True | False | False |
| True | True | True |
3.4Converse, Inverse, and Contrapositive
When there is a proposition in the form of “”, “” with and reversed is called the “converse” proposition. Also, “” with negation added to and is called the “inverse”, and “” which is both converse and inverse is called the “contrapositive”.
”, “” with and reversed is called the “converse” proposition. Also, “” with negation added to and is called the “inverse”, and “” which is both converse and inverse is called the “contrapositive”.
Among these, the contrapositive is particularly important, and the contrapositive is equivalent to the original proposition. For example, for the proposition “if , then is an odd number”, the contrapositive is “if is not an odd number, then it is not ”, and these two propositions are equivalent.
, then is an odd number”, the contrapositive is “if is not an odd number, then it is not ”, and these two propositions are equivalent.In other words, when you want to prove a proposition, you can prove the original proposition by proving the contrapositive proposition instead of proving the original proposition.
3.5De Morgan’s Laws
Also, as an important law, there are “De Morgan’s laws”.
De Morgan’s laws are the laws that “” and “” are equivalent, and “” and “” are equivalent. To break it down, it is a law that when the parentheses of “
” are removed, the “” and “” inside are swapped, and “” is distributed.
” and “” are equivalent, and “” and “” are equivalent. To break it down, it is a law that when the parentheses of “” are removed, the “” and “” inside are swapped, and “” is distributed.For example, the proposition “it is not ‘ is an even number and is 
or more’” is the same as saying “ is not an even number, or is not or more”. Also, “it is not ‘ is an even number or is or more’” is the same as saying “ is not an even number, and is not or more”.
is an even number and is or more’” is the same as saying “ is not an even number, or is not or more”. Also, “it is not ‘ is an even number or is or more’” is the same as saying “ is not an even number, and is not or more”.It is useful when you want to transform and organize complex propositions.
4.Propositional Functions
To handle a wider variety of theorems and propositions, let’s delve a little deeper into logical formulas.
Something that becomes a proposition when it receives a value from the outside is called a “propositional function”. For example, for the description “it is 
”, if you substitute for and 
for , it becomes the proposition “it is ”, so “it is ” is a propositional function.
”, if you substitute for and for , it becomes the proposition “it is ”, so “it is ” is a propositional function.
In addition to specific values such as “” and “”, propositional functions can take things like “all values” and “some value”. By adding the symbols “
” and “
” before letters such as “
” and “
”, they represent “all values” and “there exists some value”, respectively.
” and “”, propositional functions can take things like “all values” and “some value”. By adding the symbols “” and “” before letters such as “” and “”, they represent “all values” and “there exists some value”, respectively.For example, if you enclose the propositional function “it is ” with and substitute for and write it as “it is ”, it represents the proposition “for all values , it is ”. Similarly, if you enclose it with and substitute for and write it as “it is ”, it becomes the proposition “there exists some value such that it is ”.
” with and substitute for and write it as “it is ”, it represents the proposition “for all values , it is ”. Similarly, if you enclose it with and substitute for and write it as “it is ”, it becomes the proposition “there exists some value such that it is ”.
As a specific example, suppose there is a propositional function “”, and the proposition “” with substituted for and is True, and the proposition “” with substituted for and for is False.
”, and the proposition “” with substituted for and is True, and the proposition “” with substituted for and for is False.At this time, because there is “”, “” does not become True for all and . Therefore, “” is False. Also, because there is “”, there exist at least some and such that “” becomes True. Therefore, “” is True.
”, “” does not become True for all and . Therefore, “” is False. Also, because there is “”, there exist at least some and such that “” becomes True. Therefore, “” is True.5.Intuitionistic Logic
Finally, I will briefly introduce a different way of thinking called “intuitionistic logic”.
Until now, we have assumed the “law of excluded middle”, which states that when there are propositions and , at least one of them is a theorem, but intuitionistic logic does not use this law of excluded middle. In other words, with the logic so far, we could say “I don’t know if you like mathematics, but you either like mathematics or you don’t”, but with intuitionistic logic, we can’t even say this, and it becomes “I don’t even know if you either like mathematics or you don’t”. It considers the possibility that we don’t know if it can be proved.
and , at least one of them is a theorem, but intuitionistic logic does not use this law of excluded middle. In other words, with the logic so far, we could say “I don’t know if you like mathematics, but you either like mathematics or you don’t”, but with intuitionistic logic, we can’t even say this, and it becomes “I don’t even know if you either like mathematics or you don’t”. It considers the possibility that we don’t know if it can be proved.If we do not assume the law of excluded middle, many theorems cannot be proved, so intuitionistic logic is not mainstream in many fields of mathematics, but it is highly compatible and often used in fields that target logic itself and computer science.
This time, we explained the basic rules of mathematics. Next time, let’s actually prove a theorem from specific axioms!
