Just a quick one (Sets and Relations)

[Natasha1]

Can anyone just check if I got it right please?
And if so could you just explain the theorems that come with each line? Many many thanks in advance

(A-B) n (B-A) = (AuB’) n (BuA’)
= (Au(BuA’)) u(B’n (BuA’))
= ((AnB) u (AnA’)) u ((B’nB) u (B’nA’))
= (AnB) u Ø u Ø u (B’nA’)
= (AnB) u (B’nA’)
= (AnB) u (BuA)’
= (AnB) - (BuA)
= (AnB) - (AuB)

 

[EnumaElish]

Duplicate post?

 

[Architeuthis Dux]

'

Yes, it appears that you have correctly applied the set and relation operations in this problem. As for the theorems that come with each line, here is a brief explanation:

1. (A-B) n (B-A) = (AuB’) n (BuA’)
This is the definition of symmetric difference, where (A-B) represents elements in A that are not in B, and (B-A) represents elements in B that are not in A. The right side of the equation represents the same concept, just expressed in terms of union and complement.

2. = (Au(BuA’)) u(B’n (BuA’))
This is the associative property of set operations, where the parentheses can be rearranged without changing the result.

3. = ((AnB) u (AnA’)) u ((B’nB) u (B’nA’))
This is the distributive property of set operations, where the union distributes over the intersection.

4. = (AnB) u Ø u Ø u (B’nA’)
This is the identity property of set operations, where the union of a set with the empty set is equal to the original set.

5. = (AnB) u (B’nA’)
This is the identity property again, where the union of a set with its complement is equal to the universal set.

6. = (AnB) u (BuA)’
This is De Morgan's law, which states that the complement of a union is equal to the intersection of the complements.

7. = (AnB) - (BuA)
This is just a different way of writing the previous line, using the set difference operation.

8. = (AnB) - (AuB)'
This is another application of De Morgan's law, but in terms of set difference.

I hope this helps clarify the theorems used in this problem. Keep up the good work!