SYMMETRIC DIFFERENCE OF TWO SETS

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}

Δ B = (A\B) u (B\A)

Δ B = {1, 3} u {0, 7}

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 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}

Δ B = (A\B) u (B\A)

Δ B = {0, 2, 7, 9} u {1, 3, 4, 7}

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

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

Related Pages

1. Union of sets

2. Intersection of sets

3. Set difference

4. Complement of a set

5. Disjoint sets

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