Is there any branch of mathematics in which it is false that
If A = B, then B = A ? ?

Question

Can the logical inference
If A = B, then B = A 
ever be false? Is there an algebra or type of logic or math entity in which this can be false?

Puzzling · Accepted Answer

The relation "is equal to" is the canonical example of a binary equivalence relation which is reflexive, symmetric and transitive.

If the equal sign (=) is being used to represent the "is equal to" relation, then it is always true that:
A = A (reflexive)
A = B → B = A (symmetric)
A = B, B = C → A = C (transitive)

But it is possible to redefine the meaning of the equal sign to where this wouldn't be true. So how are you defining '='?

Anonymous · Answer

Statement A = all dogs are 4 legged creatures.

Statement B = All 4 legged creatures are dogs.

B is not equal to A but A = B.

Logic could be such branch.

But if A and B are numbers, then it can't be false.

fcas80 · Answer

All boy children are male.
All males are not boy children.

? · Answer

I do not know everything ... anything is possible...

Is there any branch of mathematics in which it is false that If A = B, then B = A ? ?

4 Answers