Boolean Algebra

Boolean algebra. This is Logic, which discusses the truth or falsity of propositions, translated into algebra.

In order to express logical structures algebraically, we introduce a few symbols.

Every proposition is denoted by a symbol, such as $x$. For example, the proposition “Sojin Park is a person” can be denoted as $a$, and the proposition “Sojin Park is a man” can be denoted as $b$.

The AND operation, which holds only when both propositions are true, is denoted by writing them side by side. For example, the proposition “Sojin Park is a person. And Sojin Park is a man.” is denoted as $ab$.

Every proposition, when connected to itself by the AND operation, becomes itself. For example, “Sojin Park is a person. And Sojin Park is a person.” has the same truth value as “Sojin Park is a person.” Expressed symbolically, this is $aa=a$. Since this law holds for any proposition, it can also be expressed as $xx=x$.

Let us look at the equation $xx=x$ not as a proposition but simply as an equation. If $x$ is a real number, what values of $x$ satisfy this equation? It can only be $x=0$ or $1$. This matches the law of excluded middle in logic, which states that a value can only be either true or false.

If we think of $0$ as false and $1$ as true, we can see that many logical laws retain the same meaning even when expressed as equations. For example, any proposition connected to false by the AND operation becomes false. Expressed as an equation, this is $0x=0$. Also, any proposition connected to true by the AND operation becomes itself. Expressed as an equation, $1x=x$.

Boolean algebra is the study of transforming logical structures into equations in this way, in order to better understand existing logical formulas or to derive new discoveries.