As a programmer I would solve this problem as following:

Thesis:

A: B is True B: A is False 

Definition:

A = (B = TRUE) B = (A = FALSE) 

Solution:

Let's try explaining it mathematically with solving A.

A = (B = TRUE) | INSERT B A = ((A = FALSE) = TRUE) A = A = FALSE = TRUE A = FALSE = TRUE 

... but A cannot be FALSE and TRUE at the same time. I.e.

A = FALSE ≠ TRUE ⇒ A CANNOT BE SOLVED. 

If A cannot be solved, then B should also be unsolvable. Let's check:

B = (A = FALSE) | INSERT A B = ((B = TRUE) = FALSE) B = B = TRUE = FALSE B = TRUE ≠ FALSE ⇒ B CANNOT BE SOLVED. 

Conclusion:

Result 1:

1. If A is TRUE, B has to be TRUE.
2. But if B is TRUE, A has to be FALSE.
3. If A is FALSE now, then B is FALSE.
4. And if B is FALSE, A must be TRUE.

⇒ It's recursive. You have to go to the begin (1) again.

Result 2:

Imagine A and B are functions/methods that have the property "executable = TRUE|FALSE".

When you call A, B is marked as TRUE -> executable.

And when you call B, A is set to false -> not executable.

Then you could not call A anymore.

I.e. due to the order:

B = TRUE A = FALSE 

As a programmer I would concluse this problem as following:

Thesis:

A: B is True B: A is False 

Definition:

A = (B = TRUE) B = (A = FALSE) 

Solution:

Let's try explaining it mathematically with solving A.

A = (B = TRUE) | INSERT B A = ((A = FALSE) = TRUE) A = A = FALSE = TRUE A = FALSE = TRUE 

... but A cannot be FALSE and TRUE at the same time. I.e.

A = FALSE ≠ TRUE ⇒ A CANNOT BE SOLVED. 

If A cannot be solved, then B should also be unsolvable. Let's check:

B = (A = FALSE) | INSERT A B = ((B = TRUE) = FALSE) B = B = TRUE = FALSE B = TRUE ≠ FALSE ⇒ B CANNOT BE SOLVED. 

Conclusion:

1. If A is TRUE, B has to be TRUE.
2. But if B is TRUE, A has to be FALSE.
3. If A is FALSE now, then B is FALSE.
4. And if B is FALSE, A must be TRUE.

⇒ It's recursive. You have to go to the begin (1) again.

A refers to B, to its own referent. So A is referring to itself. This semantic paradox is also known as liar paradox.

A or B has to lie. Then the approach above is solvable. Or you have to spend the rest of your life searching for a mathematical solution. You won't find. ;-)