In these lessons, we will learn the complement of the intersection of sets, the symmetric difference of two sets and the symmetric difference of three sets.

The complement of the set*X* ∩ *Y* is the set of elements that are members of the universal set U but not members of *X* ∩ *Y*. It is denoted by (*X* ∩ *Y*) ’.

The symmetric difference of two sets is the collection of elements which are members of either set but not both - in other words, the union of the sets excluding their intersection. Forming the symmetric difference of two sets is simple, but forming the symmetric difference of three sets is a bit trickier.

Related Topics: More Lessons on Sets

**Symmetric Difference of two sets and three sets**
**Symmetric Difference of Sets**

Definition and properties of the symmetric difference of two sets.**Learn the difference of sets**

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The complement of the set

The symmetric difference of two sets is the collection of elements which are members of either set but not both - in other words, the union of the sets excluding their intersection. Forming the symmetric difference of two sets is simple, but forming the symmetric difference of three sets is a bit trickier.

Related Topics: More Lessons on Sets

**Example:
**

Suppose U = set of positive integers less than 10,

* X* = {1, 2, 5, 6, 7} and *Y* = {1, 3, 4, 5, 6, 8} .

a) Draw a Venn diagram to illustrate (* X* ∩ *Y* ) ’

b) Find (* X* ∩ *Y * ) ’

**Solution: **

a) First, fill in the elements for *X* ∩ *Y* = {1, 5, 6}

Fill in the other elements for *X* and *Y* and for U

Shade the region outside *X* ∩ *Y* to indicate (*X* ∩ *Y* ) ’

b) We can see from the Venn diagram that

(*X* ∩ *Y* ) ’ = {2, 3, 4, 7, 8, 9}

Or we find that *X* ∩ *Y* = {1, 5, 6} and so

(*X* ∩ *Y* ) ’ = {2, 3, 4, 7, 8, 9}

Definition and properties of the symmetric difference of two sets.

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