Proving the Equality of (A Φ B) Φ (C Φ D) and (A Φ C) Φ (B Φ D)

[majeedh]

this is another problem which i don't know how to start..im not sure which proof i should use to solve this problem
the problem is:
If A,B,C,and D are sets, does it follow
(A Φ B) Φ (C Φ D) = (A Φ C) Φ (B Φ D)
the symbol that is separting the characters is called the oplus symbol, that's the closet symbol i could find
the oplus symbol is a circle with one line going across it and one line going down

 

[HallsofIvy]

Darned if I know because I can't tell what symbol you used! Either write it out in letters or use LaTex.

 

[Architeuthis Dux]

it

I would approach this problem by first defining the oplus symbol and its properties. The oplus symbol is also known as the exclusive or (XOR) operator, which is a logical operator that returns true only when one of the operands is true. In set theory, this means that the oplus symbol represents the symmetric difference of two sets, which is the set of elements that are in either one of the sets, but not in both.

Now, to prove the equality of (A Φ B) Φ (C Φ D) and (A Φ C) Φ (B Φ D), we can use the properties of the oplus symbol and the definition of the symmetric difference. Let's start by expanding both sides of the equation:

(A Φ B) Φ (C Φ D) = (A ∪ B) Φ (C ∪ D) - (A ∩ B) Φ (C ∩ D)
(A Φ C) Φ (B Φ D) = (A ∪ C) Φ (B ∪ D) - (A ∩ C) Φ (B ∩ D)

We can see that both sides are composed of two sets being subtracted from each other. To prove that they are equal, we need to show that the two sets being subtracted are the same.

Let's take a closer look at (A ∪ B) Φ (C ∪ D) and (A ∪ C) Φ (B ∪ D). By the definition of the symmetric difference, we know that (A ∪ B) Φ (C ∪ D) is the set of elements that are in (A ∪ B) or (C ∪ D), but not in both. Similarly, (A ∪ C) Φ (B ∪ D) is the set of elements that are in (A ∪ C) or (B ∪ D), but not in both.

Now, let's focus on the elements that are in both (A ∪ B) and (C ∪ D). These elements will be present in both sets, but since we are using the oplus symbol, they will be subtracted from the final result. This means that these elements will not be included in the final result, making (A ∪ B) Φ (C ∪ D) and (