For any two two sets, the following statements are true.
(i) Union distributes over intersection :
Au(BnC) = (AUB)n(AuC)
(ii) Intersection distributes over union
An(BuC) = (AnB)u(AnC)
Example 1 :
Given :
A = {0, 1, 2, 3, 4}
B = {1, -2, 3, 4, 5, 6}
C = {2, 4, 6, 7}
Show that
Au(BnC) = (AuB)n(AuC)
Also, verify the above using Venn diagram.
Solution :
BnC = {1, -2, 3, 4, 5, 6} n {2, 4, 6, 7}.
BnC = {4, 6}
Au(BnC) = {0, 1, 2, 3, 4} u {4, 6}
Au(BnC) = {0, 1, 2, 3, 4, 6} -----(1)
AuB = {0, 1, 2, 3, 4} u {1, - 2, 3, 4, 5, 6}
= {-2, 0, 1, 2, 3, 4, 5, 6}
AuC = {0, 1, 2, 3, 4} u {2, 4, 6, 7}
= {0, 1, 2, 3, 4, 6, 7}
(AuB)n(AuC) = {-2, 0, 1, 2, 3, 4, 5, 6} n {0, 1, 2, 3, 4, 6, 7}
(AuB)n(AuC) = {0, 1, 2, 3, 4, 6} ----(2)
From (1) and (2),
Au(BnC) = (AuB)n(AuC)
Venn Diagram :
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Example 2 :
Given :
A = {x : - 3 ≤ x < 4, x ∊ R}
B = {x ; x < 5, x ∊ N}
C = {- 5, - 3, -1, 0, 1, 3}
Show that
An(BuC) = (AnB)u(AnC)
Solution :
A = {x : - 3 ≤ x < 4, x ∊ R} ----> A = {-3, -2, -1, 0, 1, 2, 3}
B = {x ; x < 5, x ∊ N} ----> B = {1, 2, 3, 4}
C = {- 5, - 3, -1, 0, 1, 3}
BuC = {1, 2, 3, 4} u {- 5, - 3, - 1, 0, 1, 3}
BuC = {-5, -3, -1, 0, 1, 2, 3, 4}
An(BuC) = {-3, -2, -1, 0, 1, 2, 3} n {-5, -3, -1, 0, 1, 2, 3, 4}
An(BuC) = {-3, -1, 0, 1, 2, 3} ----(1)
AnB = {-3, -2, -1, 0, 1, 2, 3} n {1, 2, 3, 4}
AnB = {1, 2, 3}
AnC = {-3, -2, -1, 0, 1, 2, 3} n {- 5, - 3, - 1, 0, 1, 3}
AnC = {-3, -1, 0, 1, 2, 3}
(AnB) u (AnC) = {1, 2, 3} U {-3, -1, 0, 1, 2, 3}
(AnB) u (AnC) = {-3, -1, 0, 1, 2, 3} ----(2)
From (1) and (2),
An(BuC) = (AnB)u(AnC)
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