Set Theory and Binary Logic: Understanding XOR in Set Theory Operations

[Jhenrique]

First: relating some ideia of set theory and binary logic, like:

U = 1
Ø = 0

thus, some identities appears:

U ∪ U = U
U ∪ Ø = U
Ø ∪ U = U
Ø ∪ Ø = Ø

U ∩ U = U
U ∩ Ø = Ø
Ø ∩ U = Ø
Ø ∩ Ø = Ø

1 + 1 = 1
1 + 0 = 1
0 + 1 = 1
0 + 0 = 0

1 × 1 = 1
1 × 0 = 0
0 × 1 = 0
0 × 0 = 0

So, the conclusion is that the operation of Union is analogous to AND, and the Intersection is analogous to OR.

But, one thing no is clear for me yet: and the binary operation XOR, XOR have a analogue in set theory?

 

[D H]

XOR is the same as "not equals", and sets can be compared for equality (or lack thereof).

 

[Jhenrique]

Wait... binary operations shouldn't be compared with set operations ?

 

[D H]

Wait... binary operations shouldn't be compared with set operations ?

Huh?

So, the conclusion is that the operation of Union is analogous to AND, and the Intersection is analogous to OR.

The other way around. Union is analogous to OR, intersection to AND.

But, one thing no is clear for me yet: and the binary operation XOR, XOR have a analogue in set theory?

Symmetric difference, perhaps.

Don't get too carried away with analogies. There are sixteen functions that map a pair of booleans to a boolean.

 

[Jhenrique]

I compared AND with Union and OR with Intersection. AND, OR, Union and Intersection are all operations. I think strange to compare XOR (an operation) with the ideia of "not equals" (that isn't an operation).

 

[D H]

I compared AND with Union and OR with Intersection.

And that was an erroneous comparison. Look at your own opening post. Anything AND false is false. The intersection between any set and the null set is the null set. AND is analogous to set intersection, not set union. Similarly, OR is analogous to set union, not set intersection.

AND, OR, Union and Intersection are all operations. I think strange to compare XOR (an operation) with the ideia of "not equals" (that isn't an operation).

Of course "not equals" is an operation. There's even a special symbol for it: ≠. Boolean not equals and boolean exclusive or have the exactly same truth tables. They are the same operation in boolean algebra.

 

[micromass]

I would say the equivalent to XOR is the operation

$$A\Delta B = \{x~\vert~(x\in A)~\mathrm{XOR}~(x\in B)\}$$

Thus we see easily that this is

$$A\Delta B = (A\cup B)\setminus (A\cap B)$$

This is called the symmetric difference.

 

[D H]

That's what I said in post #4.

 

[Jhenrique]

And that was an erroneous comparison. Look at your own opening post. Anything AND false is false. The intersection between any set and the null set is the null set. AND is analogous to set intersection, not set union. Similarly, OR is analogous to set union, not set intersection.

OH YEAH! I was wrong! AND is to Intersection so like OR is to Union.

This is called the symmetric difference.

"symmetric difference"... huh... very interesting!