Minimal axioms for Boolean algebra
Boolean algebra is like a game where we play with things that can be either true or false. We want to find the most basic rules that help us solve every possible problem in this game. These basic rules are called axioms.
Now, to start the game of Boolean algebra, we need some symbols that can represent true and false. Let's say we use '1' to represent true and '0' to represent false. These symbols are like pieces of a puzzle that we can fit together in different ways.
The first axiom of Boolean algebra is the identity axiom. It says that if we combine true with anything, we get true, and if we combine false with anything, we get false. In other words, 1+0=1 and 0+1=1.
The second axiom is the complement axiom. It says that for every true statement, there is a false statement and vice versa. We can write this as 1+̅1=0 and 0+̅0=1, where the bar over a symbol means "not" or "opposite".
The third axiom is the associative axiom. It says that we can group any number of true/false statements together and get the same result no matter how we group them. For example, (1+1)+1=1+(1+1)=1+1+1=3.
The fourth and final axiom is the distributive axiom. It says that we can distribute one set of statements over another set of statements using certain rules. For instance, 1×(0+1)=1 and (1×0)+(1×1)=1.
These four axioms form the foundation of Boolean algebra. With just these four rules, we can solve any problem that comes our way in this game of true and false.
Now, to start the game of Boolean algebra, we need some symbols that can represent true and false. Let's say we use '1' to represent true and '0' to represent false. These symbols are like pieces of a puzzle that we can fit together in different ways.
The first axiom of Boolean algebra is the identity axiom. It says that if we combine true with anything, we get true, and if we combine false with anything, we get false. In other words, 1+0=1 and 0+1=1.
The second axiom is the complement axiom. It says that for every true statement, there is a false statement and vice versa. We can write this as 1+̅1=0 and 0+̅0=1, where the bar over a symbol means "not" or "opposite".
The third axiom is the associative axiom. It says that we can group any number of true/false statements together and get the same result no matter how we group them. For example, (1+1)+1=1+(1+1)=1+1+1=3.
The fourth and final axiom is the distributive axiom. It says that we can distribute one set of statements over another set of statements using certain rules. For instance, 1×(0+1)=1 and (1×0)+(1×1)=1.
These four axioms form the foundation of Boolean algebra. With just these four rules, we can solve any problem that comes our way in this game of true and false.