We know that
Union of two sets:
The union of two sets A and B ,written as A ∪ B ,is the set consisting of all the elements which are either in A or in B or in both, thus, A ∪ B = {x: x∈ A or x ∈ B}
Clearly,
x ∊ A ∪ B ⇒ x ∊ A or x ∊ B, and
x ∉ A ∪ B ⇒ x ∉ A and x ∉ B.
some important results on operations on sets
A ⊆ A ∪ B , B ⊆ A ∪ B , A ∩ B ⊆ A , A ∩ B ⊆ B
It is given that A∪B = A, and B∪C = B
A∪B = A
⇒ B⊂A, and
B∪C = B
⇒ C⊂B
⇒ C⊂A
(∵ B⊂A)
⇒ A∪C = A
Therefore, Let A∪B = A and B∪C = B. Then A∪C = A.