Match the following sets for all sets A, B and C

Explanation

(i) `[(A' uu B') - A]' = [(A' uu B') nn A']' , [:' A - B = A nn B']`
`= [(A nn B)' nn A']' , [:'(A nn B)' = A' uu B']`
` = [(A nn B)']' uu (A')' = (A nn B) uu A , [:' (A')' = A]`
`= A`
(ii) `[B' uu (B' - A )]' = [B' uu (B' nn A')]' , [:' A - B = A nn B']`
`= [B' uu (B uu A)']' , [:' A' nn B' = (A uu B)']`
`= (B')' nn [(B uu A)']' , [:' (A uu B)' = A' nn B']`
` = B nn (B uu A) , [:' (A')' = A]`
`= B`
(iii) `(A - B) - (B - C) = (A nn B') - (B nn C')`
`= (A nn B') nn (B nn C')' , [:' A - B = A nn B']`
`= (A nn B') nn [B' uu (C')']`
`= (A nn B') nn (B' uu C), [:'(A')' = A]`
`= [A nn (B' uu C)] nn [B' nn (B' uu C)]`
`= [A nn (B' uuC)] nn B'`
` = (A nn B') nn [(B' uu C) nn B']`
`= (A nn B') nn B' = A nn B' = A - B`
Alternate Method
It is clear from the diagram, `(A-B) - (B-c) = A - B`

(iv) `(A - B) nn (C - B)`
`rArr (A nn B') nn (C nn B') , [:' A - B = A nn B']`
`rArr (A nn C) nn B'`
`rArr (A nn C) - B , [:' A nn B' = A - B]`
(v) `A xx B nn C = (A xx B) nn (A xx C)`
(vi) ` A xx (B uu C) = (A xx B) uu (A xx C)`
Hence , the correct maches are
`(i) harr (b), (ii) harr (c ), (iii) harr (a), (iv) harr (f), (v) harr (d), (vi) harr (e )`