Symmetric difference is one of the important operations on sets.
Let us discuss this operation in detail.
Let X and Y be two sets.
Now, we can define the following new set.
X Δ Y = (X\Y) u (Y\X)
X Δ Y is read as "X symmetric difference Y"
Now that X Δ Y contains all elements in X u Y which are not in X n Y and the figure given below illustrates this.
Ang also,
X\Y = X - Y
Y\X = Y - X
Example 1 :
Let A = {1, 3, 5, 6}, B = {0, 5, 6, 7}, find A Δ B.
Solution :
A\B = A - B
= {1, 3, 5, 6} - {0, 5, 6, 7}
= {1, 3}
B\A = B - A
= {0, 5, 6, 7} - {1, 3, 5, 6}
= {0, 7}
A Δ B = (A\B) u (B\A)
A Δ B = {1, 3} u {0, 7}
A Δ B = {1, 3, 0, 7}
Let two sets A and B be disjoint sets. That is , no common element between A and B or AnB = { } or AnB is a null set.
In such a case,
A Δ B = A u B
Example 2 :
Let A = {0, 2, 7, 9}, B = {1, 3, 4, 7}, find A Δ B. When A n B is null set, verify A Δ B = A u B.
Solution :
A\B = A - B
= {0, 2, 7, 9} - {1, 3, 4, 7}
= {0, 2, 7, 9}
B\A = B - A
= {1, 3, 4, 7} - {0, 2, 7, 9}
= {1, 3, 4, 7}
A Δ B = (A\B) u (B\A)
A Δ B = {0, 2, 7, 9} u {1, 3, 4, 7}
A Δ B = {0, 2, 7, 9, 1, 3, 4, 7} ----(1)
A n B = {0, 2, 7, 9} n {1, 3, 4, 7}
A n B = { } ----(2)
A n B is a null set.
A u B = {0, 2, 7, 9} u {1, 3, 4, 7}
A u B = {0, 2, 7, 9, 1, 3, 4, 7} ----(2)
From (1), (2) and (3), it is clear that if A n B is a null set, then
A Δ B = A u B
Note :
If A and B are not disjoint sets (that is, A n B is not a null set),
A Δ B ≠ A u B
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