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.

You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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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You can use the free Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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