Let `A = {1, 2}dot B = {1, 2, 3, 4}, C = {5, 6}`and `D = {5, 6, 7, 8}`. Verify that(i) `Axx (B nnC) = (A xxB) nn(A xxC)`. (ii) `A xxC i s a s u b s e t of B xxD`.

Explanation

L.H.S. `A times (B cap C)`
`B cap C `
`{1,2,3,4} cap {5,6}`
=`phi`
`A times (B cap C) `
`{1,2} times phi`
`phi`
R.H.S.`(A times B)cap (A times C)`
`A times B={1,2}times {1,2,3,4]`
`={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}`
`A times C ={1,2} times {5,6} `
`={(1,5),(1,6),(2,5),(2,6)}`
`(A times B) cap (A times C )`
`=phi`
L.H.S=R.H.S Hence proved.
`A times C={1,2}times {5,6} `
`={(1,5),(1,6),(2,5),(2,6)`
`B times D={1,2,3,4} times {5,6,7,8}`
`={ (1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(,8),(4,5),(4,6),(4,7),(4,8)}`
Since all the elements of set A `times` C are the elements of B `times ` D.
`A times C `is a subset of `B times D`.