Question

show that (a-b)-c=a-(b∪c) in discrete mathematics​

Answer 1

Answer:

Let U=A∪B∪C and let’s denote by X′ the complement of X with respect to U (where X is any subset of U ). In what follows, all sets considered will be subsets of U .

Then we can write X−Y=X∩Y′ , and therefore

(A−C)−(B−C)=(A∩C′)∩(B∩C′)′=(A∩C′)∩(B′∪C)=(A∩C′∩B′)∪(A∩C′∩C)=(A∩C′∩B′)∪∅=A∩C′∩B′

On the other hand,

(A−B)−C=(A∩B′)∩C′=A∩B′∩C′

Therefore we have proved that (A−B)−C=(A−C)−(B−C)

Intuitively, since (A−B)−C is what you get by removing elements of C from A−B , you can also remove elements of C from A and B prior to do the set difference.