Difference of Sets

We can say that the difference of sets is one of the important and fundamental set theory operations. Union and intersection are one of the other set theory operations in addition to the difference of sets. The difference of two sets A and B is a set that consists of all the elements of A that are actually not in B.

In this article, we will learn more about the difference of sets, their properties, uses as well as Venn diagrams, with examples.

What do you mean by Difference of Sets?

We can say that, difference of two sets A and B is the lists of all the elements that consists in set A but actually are not present in set B. The set notation used to show the difference between the two sets A and B is A − B or A ∖ B. A – B in set-builder notation is can be defined as follows:

A – B = {x / x ∈ A and x ∉ B}

A – B = is the real set that is actually obtained by removing the elements of A ∩ B from A.

B – A = is the real set that is actually obtained by removing the elements of A ∩ B from B.

How to Find the Difference of Sets?

Let’s take an example to understand clearly how to find the difference between two sets A and B.

Suppose there is two sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 9}

Now, if we want to find the difference of A – B of these two sets, just strike off all the elements that are present in A and B both.
So, A – B = {1, 2, ̶3̶,̶ ̶4̶,̶ ̶5̶} – {3̶ ,̶4̶ ,̶5̶, 6, 7, 9}

Here we found, A – B is the set of elements of A that are left. i.e., A – B = {1, 2}.

Another example- Find out the differences of sets A – B and B – A, where set A = {1, 2, 3, 4, 6} and set B = {2, 3, 5, 7, 9}

Solution:

Here, It is given that A = {1, 2, 3, 4,6} and B = {2, 3, 5, 7, 9}. Then

• A – B = {1, ̶2̶,̶ ̶3̶, 4, 6} – { ̶2̶,̶ ̶3̶, 5, 7, 9} = {1, 4, 6}
• B – A = { ̶2̶,̶ ̶3̶, 5, 7, 9} – {1, ̶2̶,̶ ̶3̶, 4} = {5, 7, 9}

Answer: A – B = {1, 4, 6} and B – A = {5, 7, 9}.

Order of Difference of Sets

How 5 – 3 is not the same as 3 – 5, so we also need to be very careful about the order, when we will go to compute the difference of sets. As we know that the difference of sets is not commutative. Its means that the result may be different if we change the order of the difference of the two sets. Therefore, for all the sets A and B, we can go to the conclusion that A – B need not to be equal to B – A.

Let’s consider the same example of two sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 9}.

Here, we found that the difference of A – B = {1, 2} and the difference of B – A = {6, 7, 9}. Therefore, it is very clear from this example that A – B ≠ B – A.

Difference of Sets Venn Diagram

Venn diagram is generally used to illustrate or find out the relationship between different sets in the form of circles or ellipses. Let’s see that, how can we use the Venn diagram to show or find out the difference of two sets C and D. In the following Venn diagram, the left crescent moon represents C – D and the right crescent moon

Also, we can say that difference of the two disjoint sets is a null set. Let’s understand this by an example.

Example of Difference of Sets by Venn Diagram

Consider two sets C = {a, b, c, d, e} and D = {a, e, f, g}. Then

C – D = {a̶, b, c, d, ̶e̶} – {a̶, ̶e̶, f, g} = {b, c, d}

D – C = {a̶, ̶e̶, f, g} – {a̶, b, c, d, ̶e̶} = {f, g}

Complement and Difference of Sets

The complement of a set A is shown by A’ or Ac and it is the difference between the sets U and A, where we can say that U is the universal set. i.e., A’ (or) Ac = U – A. This shows the set of all elements that are actually in the universal set which are not elements of set A.

Example of Complement of a Set

Let us consider the example of set A = {1, 2, 3} and U = {1, 2 ,3, 4, 6}, then here, the complement of A is,
A’ (or) Ac = {1̶,̶ ̶2̶,̶ ̶3̶, 4, 6} – { ̶ ̶1̶,̶ ̶2̶,̶ ̶3̶} = {4, 6}.

When the universal set is different, for example, U = {-3, -2, 0, 1, 2}, then the complement of A is,
A’ (or) Ac = {-3, -2, 0, 1̶,̶ ̶2̶} – {1̶,̶ ̶2̶, 3} = {-3, -2, 0}.

We need to pay attention to what universal set is considered for the difference.

Conclusion:

Based on above discussion, Difference of a Set:

We can easily say that the difference between sets A and B in this particular order, is the set of components that are actually present in set A but not in set B. The set difference of A and B is mainly denoted by symbol A – B and simply we can also write it as A minus B and at last the outcome or the result of A – B is not the same as B – A.

I believe that the above article on the set of differences will be very helpful for your understanding and exam preparation.