kindly help me with the proof of demorgan's law

For any two finite sets A and B;

(i)?(A U B)' = A' ? B' (which is a De Morgan's law of union).

(ii)?(A ? B)' = A' U B' (which is a De Morgan's law of intersection).


Proof of De Morgan?s law:?(A U B)' = A' ? B'

Let P = (A U B)' and Q = A' ? B'

Let x be an arbitrary element of P then x ? P ? x ? (A U B)'

? x ? (A U B)

? x ? A and x ? B

? x ? A' and x ? B'

? x ? A' ? B'

? x ? Q

Therefore, P ? Q ?????.. (i)

Again, let y be an arbitrary element of Q then y ? Q ? y ? A' ? B'

? y ? A' and y ? B'

? y ? A and y ? B

? y ? (A U B)

? y ? (A U B)'

? y ? P

Therefore, Q ? P ?????.. (ii)

Now combine (i) and (ii) we get; P = Q i.e. (A U B)' = A' ? B'


Proof of De Morgan?s law:?(A ? B)' = A' U B'

Let M = (A ? B)' and N = A' U B'

Let x be an arbitrary element of M then x ? M ? x ? (A ? B)'

? x ? (A ? B)

? x ? A or x ? B

? x ? A' or x ? B'

? x ? A' U B'

? x ? N

Therefore, M ? N ?????.. (i)

Again, let y be an arbitrary element of N then y ? N ? y ? A' U B'

? y ? A' or y ? B'

? y ? A or y ? B

? y ? (A ? B)

? y ? (A ? B)'

? y ? M

Therefore, N ? M ?????.. (ii)

Now combine (i) and (ii) we get; M = N i.e. (A ? B)' = A' U B'