Boolean Algebra Axioms and Duality

Discussion in 'Homework Help' started by Sasjumb, Jan 29, 2018.

1. Sasjumb Thread Starter New Member

Jan 29, 2018
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0
I am asked to name the 4 axioms of boolean algebra. And now i am confused because as far as i understand, axioms are basic statements or rules that are always true, and are used to build an understanding of a topic.

Here are the axioms that i know.

So do you guys know what my teacher is talking about when he means the 4 axioms of Boolean algebra? I hope i have not misunderstood this concept, if i have please correct me.

Also i understand duality is something that can only be used on axioms. << Is this correct?

Let me rephrase this so it's more clear.

Axioms
1. Axioms are basic rules or statements that are true because they just are or because they self prove them selves. 2. Axioms can only be proved after you have a basic understanding of a topic. 3 Axioms gives the same result no matter what you insert in variables
Duality

1. Duality can only be used on axioms because axioms always give the same result no matter what. 2 Duality is about converting something from positive to negative logic, or vise-versa. 3 XAND and NOR are dual of each other because when an output would be 1 of one of them, it would be 0 on the other.

Please tell me if there is something that i am missing or what i have misunderstood. Also another question is there a reason why u would convert something to other logic type using duality?

2. WBahn Moderator

Mar 31, 2012
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7,207
An axiom is a statement whose validity must be accepted and that can't be proven or disproven. It serves as the starting point for proving other things. What is an axiom and what is a theorem sometimes depend on what the foundation you are working from entails.

Systems like Boolean algebra can start from different things.

For instance, most people start with the definitions of the three binary operations, AND, OR, and NOT. With these definitions we can prove that things like commutativity, associativity, identity properties and the others.

But we don't HAVE to build up our algebra from those definitions. Instead, we can start with other axioms as defining how our system behaves and let the behavior of the operators flow from that.

So it really depends on what your instructor/author are choosing to do.

Have you tried Googling something like "axioms of Boolean logic"?

I have no idea what you mean by a statement such as "Duality can only be used on axioms".

I also have no idea what XAND is. But if you start with NOR, then the operation whose output is always the opposed of NOR would be OR. That's what NOR is short for -- Not OR.

Duality says that a positive-logic AND gate is a negative logic OR gate while a positive logic-OR gate is a negative-logic AND gate.

You can often simplify the logic, in terms of gate count (read that as reduced silicon die area, lower power, and faster logic) by mixing positive and negative logic.