Prove the following demorgans law: if A, B, U are sets such that A is a subset of U and B is a subset of U, then U - (A union B) = (U-A) intersection (U-B)

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- Jan 3rd 2007, 02:51 PMruproteinset theory
Prove the following demorgans law: if A, B, U are sets such that A is a subset of U and B is a subset of U, then U - (A union B) = (U-A) intersection (U-B)

- Jan 3rd 2007, 04:35 PMOReilly
- Jan 3rd 2007, 04:51 PMPlato
Using DeMorgan’s laws we have

Thus

- Jan 4th 2007, 01:27 PMSoroban
Hello, ruprotein!

It is difficult to provide a proof if we don't know what axioms and theorems you are allowed.

I will assume you are familiar with these rules . . .

. . . . . definition of set subtraction

. . . . . one of DeMorgan's Laws.

. . . . . the intersection of and a set is the set .

Quote:

Prove the following DeMorgan's law:

If are sets such that: then:

. .

The left side is: .

. . . . . . . . . . . . . from [1]

. . . . . . . . . . . . . from [2]

. . . . . . . . . . . . . from [3]

The right side is: .

. . . . . . . . . . . . . . from [1]

. . . . . . . . . . . . . . from [3]

Therefore, the two sides are equal . . .*QED*