If A = B, then B = A... Not Always True? Definition of "="

Well, if you are talking about two sets, then we define the equality $A = B $ $\iff A \subseteq B$ and $B \subseteq A$. Your friend misused the idea of equality in your example:

$$ \{y : y \text{ is Jacuzzi}\} \subseteq \{x : x \text{ is Hot Tub}\} $$ but $$\{x : x \text{ is Hot Tub}\} \not \subseteq \{y : y \text{ is Jacuzzi}\}.$$

Therefore $$\{x : x \text{ is Hot Tub}\} \not = \{y : y \text{ is Jacuzzi}\}.$$

Note that when he said

all Jacuzzi's are hot tubs but not all Hot Tubs are Jacuzzi's.

he was saying that for all Jacuzzis $ a \in \{y : y \text{ is Jacuzzi}\}$, there exists a hot tub $b \in \{x : x \text{ is Hot Tub}\} $ such that $a = b$; in other words, for every Jacuzzi, there exists a hot tub which is equal to it. However, there are hot tubs which don't have any jacuzzis equals to them. Be careful to differentiate whether you are talking about two elements of a set being equal, or the sets themselves being equal.

In this example, I could define equality between elements as those elements having the same barcode in a store.

Equivalence relations are symmetric so it is always true.

Your friend's example is an inclusion, so he was talking about $\subseteq$

The notation in most of these answers is a little heavy considering the target audience (the op and his/her friend, who are having this argument in the first place).

OP, you are correct. The mathematical sentence $a=b$ can be read forwards or backwards, no matter what $a$ or $b$ are. Likewise, you can reverse the order of their writing to $b=a$ if you like. What is said by this mathematical sentence is that $a$ and $b$ are different labels for the same thing. For example, you'll probably agree that the equation $2+2 = 4$ is true. You'll probably also agree that the left and right sides of this equation, despite looking different from one another, refer to the same thing. They both refer to $4$!

Your friend is making a very natural and common error. He's translating (almost!) identical English sentences into mathematical sentences, and finding that your reasoning about switching the order of equality is incorrect. It's easy to do!

Consider the following English sentences.

• My mother is Jane Smith.
• My mother is hungry.

It's natural to think that these will both translate into the mathematical sentences (equations):

• My mother = Jane Smith
• My mother = hungry

The first is valid, but the second is absolutely not! The second sentence suggests some strange thing along the lines that my mother is the concept of hunger itself. The thing to note is that the meaning of 'is' in the first and second English sentences, while similar, is not the same.

PS - this is why you should cringe whenever you see "mind = blown" written. It would really be more appropriate to say "mind: blown".