@**D18**: Once you get through the associative laws for ∧ and ∨ do you then discuss the distributive laws? (Or do you jump straight into De Morgan’s various Rules?)

I wondered once how far you could go with the analogies between conventional arithmetic and propositional logic. In arithmetic, we have commutative and associative laws for both addition and multiplication. In propositional logic, we have commutative and associative laws for both ∧ and ∨

In arithmetic, we then have a *distributive law of multiplication over addition* which tells us:

a * ( b + c ) = ( a * b ) + ( a * c )

Is there a similar distributive law in propositional logic? I don’t remember ever seeing such a thing mentioned in any class or textbook.

There is some tendency to see disjunction ( ∨ ) as analogous to addition, and conjunction ( ∧ ) as analogous to multiplication. So is conjunction distributive over disjunction? I.e., is

a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c )

So I played with it a bit, and figured it out, and discovered the surprise answer. See if you can work it our for yourself!

[spoiler]To my surprise, I found that conjunction and disjunction are *each distributive over the other* (suitable emoticon not found)! Thus,

a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c )

and also

a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c )

Whoda thunk it?[/spoiler]