Algebra complement
Introduction :
Sets: A set is a well defined group of objects or symbols. The objects or symbols are called the elements of the set. If an element e belongs to a set S, then this can be represented as e belongs to in S. If e does not belong to set S, it can be represented as e not belongs to in S.
The universal set (ξ) for an particular problem is the set which contains all the possible elements for that problem. That is, if there is any set under the discussion of the universal, its element will be there in that universal set.
The complement of a set A is the set of elements which are in the universal set ξ but not in A. The set is identified as A’. Notice that ξ’ = Ø and Ø’ = ξ.
For example
If ξ = { 1,2,3,4,5,6,7,8,9,10} and A = { 1,2,3,4,5}, what will be the elements in the set represented by A’ ?
A’ consists of all those elements in ξ which are not in A.
Therefore A’ = {6, 7, 8, 9, 10}.
Example problems on algebra complement:
Ex 1: Let ξ = { 5,6,7,8,9,10,11} and A = { 5,7,11}. Find A’
Solution: We have
A’ = Set of all those elements of ξ which are not in A
= {6, 8, 9, 10}.
Ex 2: Let ξ = N and A = {x | x belongs to in N, x is even}.Find A’
Solution: Here ξ = { 1,2,3,4,5,6,7,8,…….} and A = { 2,4,6,8,…….}.
Therefore A’ = Set of all those elements of ξ which are not in A.
= {1, 3, 5, 7, 9 …}
= {x | x ` in` N, x is odd}.
Ex 3: Out of 80 students who appeared in a combined test in Science and Mathematics. 64 passed in at least one subject. If 45 passed in Science and 52 in Mathematics, find:
(i) how many passed in both the subjects.
(ii) how many passed in Science only:
(iii) how many failed in Mathematics.
Solution: Let ξ be the set of all students who appeared in the test and let A and B be the sets of those who passed in Science and Mathematics respectively, Then
n (ξ) = 80, n (A) = 45, n (B) = 52 and n (A `uu` B) = 64.
(i) n (A `nn` B) = n (A) + n (B) – n (A `uu ` B) = 45 + 52 – 64 = 33.
Therefore Number of students who passed in both the subjects is 33.
So, we may Make Venn Diagram as shown.
(ii) Clearly, 12 passed in Science only.
(iii) Clearly, B’ is the set of those who failed in Mathematics,
Now, n (ξ) = n (B) + n (B’)
Therefore n (B’) = n (ξ) – n (B) = (80 – 52) = 28.
Hence, 28 failed in Mathematics.
Practice problems on algebra complement:
1. Let A = {ab, c, e, f} and B = {c, d, e, g} be two subsets of the universal set
ξ = {a, b, c, d, e, f, g, h}.
(ii) A’ (iii) B’ .
2. Use Venn-diagrams to prove: ( A `nn` B)' = A' `uu ` B'.
I like to share this Algebraic Equations with you all through my blog.
Sets: A set is a well defined group of objects or symbols. The objects or symbols are called the elements of the set. If an element e belongs to a set S, then this can be represented as e belongs to in S. If e does not belong to set S, it can be represented as e not belongs to in S.
The universal set (ξ) for an particular problem is the set which contains all the possible elements for that problem. That is, if there is any set under the discussion of the universal, its element will be there in that universal set.
The complement of a set A is the set of elements which are in the universal set ξ but not in A. The set is identified as A’. Notice that ξ’ = Ø and Ø’ = ξ.
For example
If ξ = { 1,2,3,4,5,6,7,8,9,10} and A = { 1,2,3,4,5}, what will be the elements in the set represented by A’ ?
A’ consists of all those elements in ξ which are not in A.
Therefore A’ = {6, 7, 8, 9, 10}.
Example problems on algebra complement:
Ex 1: Let ξ = { 5,6,7,8,9,10,11} and A = { 5,7,11}. Find A’
Solution: We have
A’ = Set of all those elements of ξ which are not in A
= {6, 8, 9, 10}.
Ex 2: Let ξ = N and A = {x | x belongs to in N, x is even}.Find A’
Solution: Here ξ = { 1,2,3,4,5,6,7,8,…….} and A = { 2,4,6,8,…….}.
Therefore A’ = Set of all those elements of ξ which are not in A.
= {1, 3, 5, 7, 9 …}
= {x | x ` in` N, x is odd}.
Ex 3: Out of 80 students who appeared in a combined test in Science and Mathematics. 64 passed in at least one subject. If 45 passed in Science and 52 in Mathematics, find:
(i) how many passed in both the subjects.
(ii) how many passed in Science only:
(iii) how many failed in Mathematics.
Solution: Let ξ be the set of all students who appeared in the test and let A and B be the sets of those who passed in Science and Mathematics respectively, Then
n (ξ) = 80, n (A) = 45, n (B) = 52 and n (A `uu` B) = 64.
(i) n (A `nn` B) = n (A) + n (B) – n (A `uu ` B) = 45 + 52 – 64 = 33.
Therefore Number of students who passed in both the subjects is 33.
So, we may Make Venn Diagram as shown.
(ii) Clearly, 12 passed in Science only.
(iii) Clearly, B’ is the set of those who failed in Mathematics,
Now, n (ξ) = n (B) + n (B’)
Therefore n (B’) = n (ξ) – n (B) = (80 – 52) = 28.
Hence, 28 failed in Mathematics.
Practice problems on algebra complement:
1. Let A = {ab, c, e, f} and B = {c, d, e, g} be two subsets of the universal set
ξ = {a, b, c, d, e, f, g, h}.
(ii) A’ (iii) B’ .
2. Use Venn-diagrams to prove: ( A `nn` B)' = A' `uu ` B'.
I like to share this Algebraic Equations with you all through my blog.