Question

Verify that (AUB)’=A’n B’

Answer 1

Assuming
`A'` denotes the complement of the set
`A`, i.e. all elements in the universal set that do not belong to
`A`, then we can prove this in general.

To establish equality between two sets, you need to show that they are both subsets of one another.

• Prove
  `(A\cup B)'\subseteq A'\cap B'`:

Let
`x\in(A\cup B)'`.

By definition of set complement, this means
`x\\ot\in A\cup B`.

By definition of set union,
`x\\ot\in A` and
`x\\ot\in B`.

By definition of complement,
`x\in A'` and
`x\in B'`.

By definition of set intersection,
`x\in A'\cap B'`.

Therefore
`(A\cup B)'` is a subset of
`A'\cap B'`, because membership of some arbitrary element in the first set directly implies membership in the second set.

• Prove
  `A'\cap B'\subseteq(A\cup B)'`:

Let
`x\in A'\cap B'`. The proof follows similarly as above.

By definition of intersection,
`x\in A'` and
`x\in B'`.

By definition of complement,
`x\\ot\in A` and
`x\\ot\in B`.

By definition of union,
`x\\ot\in(A\cup B)`.

By definition of complement,
`x\in(A\cup B)'`.

Therefore
`A'\cap B'` is a subset of
`(A\cup B)'`.

And hence both sets are equal, regardless of what the sets may be.