If A = B, then B = A... Not Always True? Definition of "="
A friend and I recently got into a silly argument where I stated A = B so B = A. He stated this was not always true. After asking for an example he stated
Jacuzzi = Hot Tub
Hot Tub ≠ Jacuzzi
Meaning all Jacuzzi's are hot tubs but not all Hot Tubs are Jacuzzi's.
Understanding that we are not completely on the same page I tried to describe the difference between our definitions of using the '=' sign but failed.
In math, what has to be true for the "=" sign to apply?
Solution 1:
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.
Solution 2:
Equivalence relations are symmetric so it is always true.
Your friend's example is an inclusion, so he was talking about $\subseteq$
Solution 3:
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".
Solution 4:
As far as math goes, "=" essentially means "is"; that they are the same forward and backwards. We can switch places: $a=b, b=a$ and that's all the is essentially needed.
Your friend is referring to ⊂, which is way different than $=$. ⊂ means a subset. Thus a hot tub is a subset of the jacuzzi set, however the entire jacuzzi set is not in the hot tub set.
$=$, when referring to sets, means that each elements of a set are contained in the other set, and have no additional elements.
For example: Let $A = (1,2,3,4,5,6)$ and $B = (1,3,5)$ This sets are entirely different, however set $B$ exists in $A$, but $A$ does not exists in $B$. Thus $A \not = B$. And in order for the equal sign to work, $A ⊂ B$ and $B ⊂ A$
Solution 5:
When you write
$$ \textrm{'Jacuzzi'} = \textrm{'Hot Tub'} $$
you already make an incorrect statement, because they are not pure identical. You should write something like
$$ \textrm{'Jacuzzi'} := \textrm{'Hot Tub'} $$
A $\textrm{'Jacuzzi'}$ is 'defined' as a $\textrm{'Hot Tub'}$
Think of 'horse' and 'animal'
We can define a horse as an animal (including all other properties), thus
$$ \textrm{'Horse'} := \textrm{'Animal'} $$