if A And B are sets and AU B=AUB,then
Answers
Step-by-step explanation:
The rectangular region represents the universal set U and the circular regions the subsets A and B. The shaded portion represents the set name below the diagram.
Let A and B be the two sets. The union of A and B is the set of all those elements which belong either to A or to B or both A and B.
Now we will use the notation A U B (which is read as ‘A union B’) to denote the union of set A and set B.
Thus, A U B = {x : x ∈ A or x ∈ B}.
Clearly, x ∈ A U B
⇒ x ∈ A or x ∈ B
Similarly, if x ∉ A U B
⇒ x ∉ A or x ∉ B
Therefore, the shaded portion in the adjoining figure represents A U B.
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Thus, we conclude from the definition of union of sets that A ⊆ A U B, B ⊆ A U B.
From the above Venn diagram the following theorems are obvious:
(i) A ∪ A = A (Idempotent theorem)
(ii) A ⋃ U = U (Theorem of ⋃) U is the universal set.
(iii) If A ⊆ B, then A ⋃ B = B
(iv) A ∪ B = B ∪ A (Commutative theorem)
(v) A ∪ ϕ = A (Theorem of identity element, is the identity of ∪)
(vi) A ⋃ A' = U (Theorem of ⋃) U is the universal set.
Notes:
A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.
Solved examples of union of sets using Venn diagram:
1. If A = {2, 5, 7} and B = {1, 2, 5, 8}. Find A U B using venn diagram.
Solution:
According to the given question we know, A = {2, 5, 7} and B = {1, 2, 5, 8}
Now let’s draw the venn diagram to find A union
Therefore, from the Venn diagram we get A U B = {1, 2, 5, 7, 8}
2. From the adjoining figure find A union B.
Solution:
According to the adjoining figure we get;
Set A = {0, 1, 3, 5, 8}
Set B = {2, 5, 8, 9}
Therefore, A union B is the set of elements which in set A or in set B or in both.
Thus, A U B = {0, 1, 2, 3, 5, 8, 9}
● Set Theory
● Sets Theory
● Representation of a Set
● Types of Sets
● Finite Sets and Infinite Sets
● Power Set
● Problems on Union of Sets
● Problems on Intersection of Sets
● Difference of two Sets
● Complement of a Set
● Problems on Complement of a Set
● Problems on Operation on Sets
● Word Problems on Sets
● Venn Diagrams in Different Situations
● Relationship in Sets using Venn Diagram
● Union of Sets using Venn Diagram
● Intersection of Sets using Venn Diagram
● Disjoint of Sets using Venn Diagram
● Difference of Sets using Venn Diagram
● Examples on Venn Diagram
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