Let A and B be sets. Prove A=B implies B=A.

Question

In order to prove the foregoing statement we use the next lemma:

For a set $A, A = A$.

Let $A$ and $B$ be a couple of sets. Assume that $A = B$. Notice we must show that $B = B$ for $A = B$. However we know by the stated lemma that $B = B$ is a true statement for an arbitrary set $B$. As a result, $B = A$ is trivially true.

Is this proof right?

Answer 1

Given that two sets are said to be equal if and only if for all elements of each set, those elements are in the other set, respectively

Then, if A=B,

it means that all elements of A are also elements of B AND all elements of B are also elements of A.

So, conversely, because AND is commutative,

all elements of B are also elements of A AND all elements of A are also elements of B,

thus B=A

Using math notation:

$ A=B \iff ((\forall a \in A \implies a \in B) \land (\forall b \in B \implies b \in A)) \iff ((\forall b \in B \implies b \in A) \land (\forall a \in A \implies a \in B)) \iff B=A $

Answer 2

I don't really follow your proof, seems like a strange exercise in casuistics. Here is a direct one instead.

• Assume $A = B$ happens.
• Therefore, $A \subseteq B$ and $B \subseteq A$.
• Hence, changing the order of these statements, $B \subseteq A$ and $A \subseteq B$.
• Thus, plugging back into the definition of equality of sets, $B = A$.

In summary, $A = B \implies B = A$. The reverse implication follows without loss of generality by exchanging $A$ and $B$.

Answer 3

The axiom of extensionality says that $$A=B \leftrightarrow (\forall x ((x\in B)\leftrightarrow (x\in A)))$$ The proof is just trivial, I don't even understand why you were told to prove this...

Answer 4

• By a truth table, one can see that

$$ (P\iff Q) \equiv (Q\iff P) $$

i.e. , that the two propositions are logically equivalent( $\equiv$), and can , therefore, be substituted one for the other.

Note : this property can be called the commutative property of the " iff " operator.

• Let $P = x\in A$ and $Q= x\in B$

• $A = B$

$\equiv \forall (x) ( x\in A \iff x\in B)$ ( By the extensionality principle defining set equality )

$\equiv \forall (x) ( x\in B \iff x\in A)$ ( By the commutativity property if " iff" )

$\equiv \ B = A $ ( Again, by the extensionality principle).

• Therefore $A = B \equiv B = A $ ( By transitivity of logical equivalence).

• But two equivalent propositions logically imply each other ( equivalence is reciprocal implication, which can be checked by a truth table). So, in particular,

$$A = B \implies B = A$$