Complement of a Set: Class 11 Maths Notes, Definition, Properties & Solved Examples

The complement of a set A is the set of all elements in the universal set $U$ that are not in A. One way to visualise this complement meaning in math is to see a Venn diagram and know that it supports the set A to form a whole of the universal set. It gives us everything that is outside the set A, which would not include elements within it but outside it under U or universal set. This is the basic complement of a set definition.

But we also need to see how to represent this in maths. We could easily show that as $A^{\text{'}}=\{x\in U:x\notin A\}$ . The other form is a little simpler to show, like this $A^{\wedge }c=U-A$

Approaching the Complement of a Set for Exams

1. The questions for complement of a set in JEE Mains can be both basic or a little intermediate, but it's ideal to prepare for the tougher ones. You can expect questions on the direct application of the definition, its properties, and the integrated set complements with functions and other topics like probability. 
2. In previous years' questions for JEE, De Morgan's Laws and visualising sets have been common.  Meanwhile, you can brush up on your knowledge of the entire chapter with our detailed guide on sets and their representations.

In Chapter 1 of your NCERT Maths textbook, the definition of the Complement of a Set is in Section 1.10. It tells,

"Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A. Symbolically, we write A′ to denote the complement of A with respect to U.

Thus,
A′ = {x : x ∈ U and x ∉ A }. Obviously A' = U – A"

See it like this. 

U is the full collection of items you're working with in a whole set. That's the universal set.

Considering now that A is a smaller set inside U, we will see that A only has some of the elements.

Now the complement of set A is represented as A′.  That is everything in U. And it's not in A then. 

Simply said, if it’s in U but not in A, it goes somewhere else. That is why we know it's in A′.

So in this case, you can write in the formulaic manner. 

A′ = {x : x is in U but not in A}

Or even more simply:
A′ = U – A

 

Class 11 Set Theory from your NCERT book covers 4 properties, which are great when preparing before your competitive tests after boards.

Try to remember these for the numerical problems, while you'll be good to go in your exams. One tip here would be to immediately test your knowledge of these with real numerical problems. They'll be really helpful.

• Representing the Double Complement Law as A''=A

• Universal Set and Empty Set Laws are seen in two ways, one which is U'=∅ and the other is ∅'=U
• De Morgan's Law would be shown as (A ∪ B)´ = A′ ∩ B′ and the other way to show is (A ∩ B)′ = A′ ∪ B′
• Complement Laws, which are A ∪ A′ = U and A ∩ A′ = φ

In the two important representations below  

• (A∪B)'=A'∩B' - this is the complement of a union, telling us that everything not in A or B is the same as everything that’s in neither A nor B.
• (A∩B)'=A'∪B', where everything not in both A and B is the same as everything that’s either not in A or not in B. This is the complement of intersection.

We see that these are how the Laws of De Morgan are represented when we are learning the Set Theory.

Example 1:
Let $\mathrm{U}=\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6},\mathrm{7,8}\}$ and $\mathrm{A}=\{\mathrm{2,4},\mathrm{6,8}\}$

Then ${\mathrm{A}}^{\text{'}}=\{\mathrm{1,3},\mathrm{5,7}\}$

 

Example 2:

If $\mathrm{U}=$  natural numbers less than 10 and $\mathrm{B}=\{\mathrm{1,3},\mathrm{5,7},9\}$ , then ${\mathrm{B}}^{\text{'}}=\{\mathrm{2,4},\mathrm{6,8}\}$

The complement of sets is useful in real-life scenarios like survey results, student enrolments, etc.

For example, if a group of students in a class are currently enrolled in the Math subject as an elective is set A, and all students are in U. The ones who are not in the Math subject are in A'.

Another simple example of a complement of a set is to consider an if-then scenario. If $60\mathrm{\%}$  like tea (A), $40\mathrm{\%}$ are in A' (the ones who do not like it).

If we look at a more intermediate level understanding of set complements, we need to know about multiple sets.   

Let $A=\{\mathrm{1,2},3\},B=\{\mathrm{3,4},5\}$ , and $U=\{\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\}$
$A\cup B=\{\mathrm{1,2},\mathrm{3,4},5\}$ , $(A\cup B{)}^{\text{'}}=\{6\}$
$A^{\text{'}}=\{\mathrm{4,5},6\},{\mathrm{B}}^{\text{'}}=\{\mathrm{1,2},6\},{\mathrm{A}}^{\text{'}}\cap {\mathrm{B}}^{\text{'}}=\left\{6\right\}$

So, $(\mathrm{A}\cup \mathrm{B}{)}^{\text{'}}={\mathrm{A}}^{\text{'}}\cap {\mathrm{B}}^{\text{'}}$ ,