मान लीजिए कि A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} तथा D = {5, 6, 7, 8} । सत्यापित कीजिए कि-
(i) `A xx (B nn C) = (A xx B) nn (A xx C)`
(ii) `A xx C, B xx D` का एक समुच्चय है ।

Explanation

(i) `because A = {1, 2}, B = {1, 2, 3, 4} ` और C = {5, 6}
`therefore B nn C` = समुच्चय B व C के उभयनिष्ठ अवयवों का समुच्चय `phi`
`therefore A xx (B nn C) = A xx phi = phi`
`A xx (B nnC) = phi " "… (i)`
अब `A xx B` = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
और `A xx C = {(1, 5), (1, 6), (2, 5), (2, 6)}`
`therefore (A xx B) nn (A xx C) = (A xx B)` व`(A xx C)` के सर्वनिष्ठ क्रमित-युग्मों का समुच्चय `= phi`
`therefore (A xx B) nn (A xx C) = phi " " ... (2)`
तब समीकरण (1) व (2) से
`A xx (B nnC) = (A xx B) nn (A xx C)` यही सिद्ध करना था ।
(ii) `because A` = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6},
D = {5, 6, 7, 8}
तब `A xx C` = {(1, 5), (1, 6), (2, 5), (2, 6)} ... (1)
और `B xx D`= {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
स्पष्ट है कि `A xx C` के सभी क्रमित युग्म `B xx D` में है ।
अतः `A xx C, B xx D` का एक उपसमुच्चय है ।