The different meanings of equality in Mathematics

 At the ripe age of 50, I have started understanding the "=" sign in math.

For some reason, I never noticed this sign for so long. I thought it was simple - it meant a number or variable is equal to another.

But recently, trying to read George Boole's Mathematical Analysis of Thought, I understood that "=" can mean different thing is different contexts.

In the realm of quantities, of course "=" means two different quantities are the same.

But in set theory, it means something different.

For example, if you have two sets, X and Y, X=Y means that every item in X is also in Y and Y has no more items.

When applied to actions, xy = yx means that the action of an operation y, followed by the action of an operation x is equal to the reverse.

If there is a basket full of different colored balls, say x is the action of selecting blue colored balls; and y is the action of selecting red colored balls, then x+y = y+x means that:

- selecting red colored balls from the basket; and then selecting blue colored balls from the basket; and

- selecting blue colored balls from the basket and then selecting red colored balls;

leaves you with the exact same collection of balls. In this case x and y are not quantities of objects but actions.

The same "=" can be applied to statements that have true or false values, called propositions.

A proposition, Today is Friday is true on all fridays and false on all other days.

A proposition Tomorrow is Saturday is also true for all all fridays and false on all other days.

Therefore, Today is Friday = Tomorrow is Saturday. The "=" sign here does not just compare true or false values - it matches only if: - all the days on which the left hand side is true are the same as all the days on which the right hand side is true; and

- all the days on which the LHS is false are the same as all the days on which the RHS is false.

This is useful in Artificial Intelligence, because often machines need to make decisions about if two different statements in a context are same or not.

So, the humble "=" sign seems to be used with entirely different meanings in different mathematical contexts.