Sets is one of the integral parts of class 11 mathematics. It introduces the set theory which is one of the basics for higher studies. Hence, the chapter is important and mustn't be neglected. The chapter is also slightly formula based and hence sets of formulas class 11 are necessary. For this reason, here we are presenting to you all the important sets of class 11 Formulas. We will be discussing all the important formulas and with that, we'll also see the significance of each of the formulas.
A set is a defined collection of objects. A set that contains a definite number of objects is called a definite set. Whereas a set consisting of an indefinite number of elements is called indefinite sets.
Example of a finite set : {1,2,3,4}.
Example of an infinite set: set of all the natural numbers.
Now to understand all the formulas, firstly let us understand all the symbols used and what they signify.
Symbol | Meaning |
N | Set of all natural numbers. |
Z | Set of all integers. |
Q | Set of all rational numbers. |
R | Set of all real numbers. |
Z+ | Set of all positive integers. |
Q+ | Set of all positive rational numbers. |
R+ | Set of all positive real numbers. |
When you want to unify or add two sets A and B, it is represented through A U B. Finding the union of two sets gives us a set containing all the elements contained in both A and B.
When you want to find the common elements between two sets A and B then, you need to find the set A intersection B or A inverted U B.
Sets can be added and subtracted.
You can also find A bar, this shall give you all the elements which are not contained in the set. This is called the complement of the set.
The cardinality of a set can be defined as the number of elements contained in a set. It could range from 0 to infinity.
For instance
Consider the set A = {1,2,3,4}.
The cardinality of the set A is represented as n(A), which is 4 since A contains 4 elements.
Let's take another example for a better understanding:
Now consider the set of all the integers Z.
What would the cardinality of Z be?
Well, we do not know the number of integers hence the cardinality of the set Z would be indefinite.
Consider two sets of unknown cardinality:
A and B.
n(AᴜB) represents the total number of elements present in both of the sets A and B combined.
n(AᴜB) = n(A) + (n(B) – n(A∩B).
The Above-mentioned Formula is one of the most important formulae of set theory. It is used to find AᴜB. Let's understand this Formula in detail using a Venn diagram.
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The diagram given above is called the Venn diagram. It represents two different sets A and B. The region of the Venn diagram which is highlighted in pink are the elements that are common to both sets A and B. If we simply add the elements of A and B together, the elements belonging to the pink part would be added twice and hence will give us an incorrect sum. Hence while finding the union of two sets we need to subtract the intersection of the two sets one time. This will negate the error caused earlier and find the perfect union between the two sets.
Now, let us jump to the next level and try understanding the formula for 3 different sets:
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Consider 3 sets intersecting like in the Venn diagram given above.
n(AUBUC) = n(A) + n(B) + n(C) – n(A⋂B) – n(B⋂C) – n(C⋂A) + n(A⋂B⋂C)
From the image, we can visualize that if we simply add the sets together, some of the regions will be added multiple times. Hence, using this Formula we rectify the error and subtract the regions that have been added multiple times.
Example 1: In a school, there are 200 children, 65 like drawing and 85 like music. 25 like both. Find out how many of them like either of them or neither of them?
Solution:
Total number of children, n(μ) = 200
Number of children who enjoy drawing, n(d) = 65
Number of children who enjoy music, n(m) = 85
Number of students who like both, n(d∩m) = 25
Number of students who like either of them,
n(dᴜm) = n(d) + n(m) – n(d∩m)
→ 65 + 85 - 25 = 125
Number of students who like neither = n(μ) – n(dᴜm) = 200 – 150 = 50.
1. Can I get a List of all Sets of Chapter Class 11 Formulas?
Ans: Given below is the formula list of all the formulae of the chapter sets class 11.
n(AᴜB) = n(A) + (n(B) – n(A∩B).
n(AUBUC) = n(A) + n(B) + n(C) – n(A⋂B) – n(B⋂C) - n(C⋂A) + n(A⋂B⋂C).
If n(A∩B)= 0 then AᴜB = n(A) + n(B).
n(A - B) + n(A∩B) = n(A).
N(B - A) + n(A∩B) = n(B).
n(AᴜB) = n(A - B) + n(A⋂B) + n(B - A).
The formulas given above are all you need for solving almost all the types of problems that you would encounter related to set theory. Knowing these formulas would enable you to solve even the toughest questions.
2. Why do you Need the Set Formula List Class 11?
Ans: The formula list given above consists of all the formulae needed to solve any kind of problem from the chapter set theory. Remembering these formulas is very vital since the chapter is highly formula based. Although you needn't mug up all the formulae, you must understand each formula from their Venn diagram. You must also understand how the formula was derived from the Venn diagram. However, additionally remembering the formulae would make it easier for you to solve the problems at a quicker pace since you cannot put in effort into deriving the formula every single time!
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