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Difference of Sets: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Difference of SetsDifference of Sets: Definition and ExamplesTable of ContentsDifference of Sets Definition of Difference of Sets

The difference between two sets, AAA and BBB, written as A∖BA \setminus BA∖B or A−BA - BA−B, is a set that contains those elements of A that are NOT in B. To find the difference, we remove all the elements of set B from set A. We can define the difference between two sets using the set builder notation as follows: A−B={x:xinA and x∉B}A-B= \{x : x \in A \text{ and } x \notin B\}A−B={x:xinA and xin/B}. Similarly, B−A={x:xinB and x∉A}B-A= \{x : x \in B \text{ and } x \notin A\}B−A={x:xinB and xin/A}. It's important to note that changing the order of the difference between two sets can lead to different results, making set difference a non-commutative operation.

There are various types and properties of set differences. For disjoint sets A and B with no common elements, A−B=AA - B = AA−B=A and B−A=BB - A = BB−A=B. The complement of a set A, denoted by A′A'A′ or AcA^cAc, is the difference between the universal set U and A, written as Ac=U−AA^c = U - AAc=U−A. Another important concept is the symmetric difference of sets, denoted as A△BA \triangle BA△B, which gives elements present in either set but not in their intersection. It can be calculated as A△B=(A−B)∪(B−A)A \triangle B = (A - B) \cup (B - A)A△B=(A−B)∪(B−A) or as A△B=(A∪B)−(A∩B)A \triangle B = (A \cup B) - (A \cap B)A△B=(A∪B)−(A∩B).

Examples of Difference of Sets Example 1: Finding the Difference Between Two Number Sets Problem:

Given set AAA = {1, 2, 3, 4, 5} and set BBB = {3, 4, 5, 6, 7}. Find the difference between sets AAA and BBB.

Step-by-step solution:

Step 1, Look at the two sets we have: AAA = {1, 2, 3, 4, 5} and BBB = {3, 4, 5, 6, 7}.

Step 2, To find A–BA – BA–B, we need to keep only the elements from set AAA that are not in set BBB.

Step 3, The elements 3, 4, and 5 appear in both sets. So we need to remove these from set AAA.

Step 4, After removing the common elements, we're left with only 1 and 2 from set AAA.

Step 5, So the set difference A–BA – BA–B = {1, 2}.

Example 2: Finding Set Difference Using Set Notation Problem:

If XXX = {a, b, c, d, e} and YYY = {c, d, e}, find XXX \ YYY.

Step-by-step solution:

Step 1, Look at our sets: XXX = {a, b, c, d, e} and YYY = {c, d, e}.

Step 2, Remember that XXX \ YYY means X–YX – YX–Y, which is the set of elements in set XXX that are not in set YYY.

Step 3, The elements c, d, and e appear in both sets, so we need to remove these from set XXX.

Step 4, After removing the common elements from set XXX, we're left with a and b.

Step 5, Therefore, XXX \ YYY = {a, b}.

Example 3: Finding Difference With Word Elements Problem:

What is the set difference (P–QP – QP–Q) between Set PPP = {apple, banana, orange, pineapple} and Set QQQ = {banana, pineapple}?

Step-by-step solution:

Step 1, Start by listing our sets: PPP = {apple, banana, orange, pineapple} and QQQ = {banana, pineapple}.

Step 2, To find P–QP – QP–Q, we need to keep only the elements from set PPP that are not in set QQQ.

Step 3, The elements "banana" and "pineapple" are in both sets PPP and QQQ, so we need to remove these from set PPP.

Step 4, After removing the common elements from set PPP, we're left with "apple" and "orange".

Step 5, So the set difference P–QP – QP–Q = {apple, orange}.

Comments(4)CCounselorTaraNovember 5, 2025I've used this glossary page to teach set differences. It's super clear with great examples. Helped my students grasp the concept easily!

GGuardXenaNovember 4, 2025I've used this diff. of sets def. with my students. It's super clear & the examples really helped them grasp the concept. Great resource!

LLeatherworkerKimNovember 4, 2025I've used this difference of sets def for my students. It's super clear with great examples. Really helped them grasp the concept!

NNatureLover82September 17, 2025I’ve been helping my kids with set theory, and this explanation of the difference of sets was super clear! The examples made it easy to understand and apply to their homework.

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