How can a logical approach help prove Boolean algebra properties?

[seang]

For my digital logic class, we are supposed to prove all of the two and three variable properties (commutative, associative, distributive...). I'm not really sure how to go about this, because although he didn't say, it seems intuitive to prove them in order, ie you can't use a property until you've proved it.

So I'm thinking about using the Single Variable Theorems for the first few (x+x = x , x*x = x , etc)

So then I started to say, for the commutative property, yx=xy, multiply each side by x to get xyx=xxy, but then, wouldn't I have to use the associatve property to group the x's together to obtain 1?, so that y=y?

Help?

 

[Hurkyl]

Well, you have to start at some definition (or list of assumed axioms) before you can prove anything; where is your starting point for + and *?

 

[Architeuthis Dux]

As a scientist, it is important to approach any problem or task with a logical and systematic approach. In this case, proving the properties of Boolean algebra requires a similar approach. It is important to understand and prove each property in order to build a solid foundation for more complex proofs.

One way to approach proving the properties is to start with the basic single variable theorems, as you mentioned. This will help establish a foundation for the more complex properties. However, it is important to note that the single variable theorems are not enough to prove all of the properties.

For the commutative property, your approach is correct. Multiplying both sides by x will result in xyx=xxy. However, instead of using the associative property to group the x's together, you can use the single variable theorem for multiplication, which states that x*x = x. This will simplify the equation to xy=x, which is the commutative property.

It is also important to note that for the distributive property, you will need to use both the single variable theorems for addition and multiplication to prove it. This is because the distributive property involves both addition and multiplication.

In conclusion, it is important to have a clear understanding of the single variable theorems and how they can be used to prove the properties of Boolean algebra. It is also important to approach each property systematically and logically, using the appropriate theorems and properties to build upon each other. With a careful and thorough approach, you will be able to successfully prove all of the two and three variable properties in your digital logic class.