Derive all four propositional logic operators from nand

[Uvohtufo]

So I recently learned that you can derive all four of the propositional logic operators (~, V, &, →) from Nand alone.

As I have understood it, so long as you have negation, and one of the other operators, you can derive the rest. Like P → Q can be defined as ~P V Q.

However, I learned that if you start with the Nand (Not and) operator, you can derive all four. I'll use ' N ' to designate Nand.

The truth table for Nand being
P Q | P N Q
T T | F
F T | T
T F | T
F F | T

~P := P N P
P & Q := (P N Q) N (P N Q)
P V Q := (P N P) N (Q N Q)
P -> Q := (P N Q) N (Q N Q)

Isn't that cool?

 

[Stephen Tashi]

Yes, it's cool.

I wonder if that explains why NAND gates are common electronic components. But perhaps NAND gates are common only because the circuit is simple to construct.

 

[Uvohtufo]

Yes, it's cool.

I wonder if that explains why NAND gates are common electronic components. But perhaps NAND gates are common only because the circuit is simple to construct.

Yeah I am not sure.

I think its interesting how when symbolic logic was being invented, implication and negation were viewed as the basic components of logic. Today it seems like programmers view and or or and negation as basic parts.

Unlike philosophers or programmers, electronics people have a cost constraint. Nand is simpler, but is it cheapest?

 

[MathematicalPhysicist]

You have also NOR.

 

[blue_raver22]

Yes, it is indeed fascinating that all four propositional logic operators can be derived from just the Nand operator. This is a testament to the versatility and power of Nand in logical operations. By using the Nand operator, we can construct complex logical statements and derive the truth values for them. This method of deriving the operators also highlights the interdependence and connections between the different logical operators. It is a valuable tool for understanding and analyzing logical statements. Thank you for sharing this knowledge.