Complementary Set
Given the universal set U, the complementary set of the set X is the set containing only those members of U that do not belong to the set X it is denoted by X’ or Xc which can be written as “complement of X”.
We take all the members of set U which does not belong to the set X, thus x ϵ X’ if x ∉ X and x ϵ U
Example – 1
Let the universal set U is the member of the natural whole number
If, the universal set U = { 1, 2, 3, 4, 5 } and X = { 3, 4, 5 } , here X’ = { 1, 2 }
Example – 2
If, X = { x / x is the consonant }
Then, X’ = { x / x is vowel }
Example – 3
If, U = { 11, 12, 13, 14, 15, ……….., 19, 20 }, A = { x / 5< x <10 }, and B = { x / 100 < x² < 225 } then find A’ & B’
Ans.) Writing both the set in tabular form A = { 6, 7, 8, 9 }, B = { 11, 12, 13, 14 }
A’ = the set of elements of U that do not belong to A = { 6, 7, 8, 9 }, but we can observe here no member of U is not available to A, so A’ = φ ( we can say, it is disjoint set)
B’ = the set of elements of U that do not belong to B = { 11, 12, 13, 14 }, we can find here B’ = { 15, 16, 17, 18, 19, 20 }.
Example – 4
If, U = { 11, 12, 13, 14, 15, ……….., 19, 20 }, A = { x / 5< x <10 } and B = { x / 100 < x² < 225 } then find A’ ⋃ B’
Ans.) Writing both the set in tabular form A = { 6, 7, 8, 9 }, B = { 11, 12, 13, 14 }
A’ = the set of elements of U that do not belong to A = { 6, 7, 8, 9 }, but we can observe here no member of U is available to A, so A’ = φ ( we can say, it is disjoint set)
B’ = the set of elements of U that do not belong to B = { 11, 12, 13, 14 }, we can find here B’ = { 15, 16, 17, 18, 19, 20 } .
So, A’ ⋃ B’ = φ U { 15, 16, 17, 18, 19, 20 } = { 15, 16, 17, 18, 19, 20 }
Example – 5
If , U = { 11, 12, 13, 14, 15, ……….., 19, 20 }, A = { x / 5< x <10 } and B = { x / 100 < x² < 225 } then find A’ ⋂ B’
Ans.) Writing both the set in tabular form A = { 6, 7, 8, 9 }, B = { 11, 12, 13, 14 }
A’ = the set of elements of U that do not belong to A = { 6, 7, 8, 9 } , but we can observe here no member of U is available to A, so A’ = φ ( we can say, it is disjoint set)
B’ = the set of elements of U that do not belong to B = { 11, 12, 13, 14 }, we can find here B’ = { 15, 16, 17, 18, 19, 20 }.
So, A’ ⋂ B’ = φ ⋂ { 11, 12, 13, 14 } = φ
Example – 6
If , U = { 11, 12, 13, 14, 15, ……….., 19, 20 }, A = { x / 5< x <10 } and B = { x / 100 < x² < 225 } then find (A ⋃ B)’
Ans.) Writing both the set in tabular form A = { 6, 7, 8, 9 }, B = { 11, 12, 13, 14 }
A ⋃ B = { 6, 7, 8, 9, 11, 12, 13, 14 }
(A ⋃ B)’ = the set of elements of U that do not belong to A ⋃ B = { 6, 7, 8, 9, 11, 12, 13, 14 }, but we can observe here no member of U is available to A ⋃ B, so (A ⋃ B)’ = φ ( we can say, it is disjoint set)
Example – 7
If, U = { 11, 12, 13, 14, 15, ……….., 19, 20 }, A = { x / 5< x <10 }, and B = { x / 100 < x² < 225 } then find (A ⋂ B)’
Ans.) Writing both the set in tabular form A = { 6, 7, 8, 9 }, B = { 11, 12, 13, 14 }
A ⋂ B = { 6, 7, 8, 9} ⋂ {11, 12, 13, 14 } = φ ( it is disjoint set)
(A ⋂ B)’ = the set of elements of U that do not belong to A ⋂ B = φ, but we can observe here no member of U is available to A ⋂ B, so (A ⋂ B)’ = φ ( we can say, it is disjoint set)