Hey so a bit of math here.

So I want to prove: (A U B)’ = A’ ∩ B’ —> De Morgan’s law.

*General case: *In set theory, at least how I remember it is, if A = B then A is a subset of B and B a subset of A. Which is really obviously true. In mathy terms A ⊆ B and B ⊆ A. (⊆ means subset).

So what we want to do here is prove that (A U B)’ ⊆ A’ ∩ B’ and then prove also A’ ∩ B’ ⊆ (A U B)’ – just reversed the order.

**STEP 1** – (A U B)’ ⊆ A’ ∩ B’

*General case:* The prove that A ⊆ B (A is a subset of B) we show that there is a value common to both A and B.

suppose x ∈ (A U B)’ (∈ means ‘is contained in’). This wuld mean that x ∉ A U B and hence x ∉ A and x ∉ B (draw a Venn diagram if this is not too clear). which leads us to x ∈ A’ and x ∈ B’when combined is x ∈ A’ ∩ B’

And hence we have proven that x is in both A’ ∩ B’ and (A U B)’ which means (A U B)’ ⊆ A’ ∩ B’

**STEP 2** – A’ ∩ B’ ⊆ (A U B)’

*Try this on your own.*

……………………

and finally (A U B)’ = A’ ∩ B’

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