# Sets: Some Properties of Operation of Intersection and Important Equation (For CBSE, ICSE, IAS, NET, NRA 2022)

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## What Are Sets?

• Sets are defined as a well-defined collection of objects.
• First, we specify a common property among “things” (we define this word later) and then we gather up all the “things” that have this common property.
• For example, the items you wear: hat, shirt, jacket, pants, and so on.
• This is known as a set.
• A set without any element is termed as an empty set.
• A set comprising of definite elements is termed as a finite set whereas if the set has an indefinite number of elements it is termed an infinite set.
• There is a simple notation for sets. We simply list each element (or “member” ) separated by a comma, and then put some curly brackets around the whole thing:
• At the start we used the word “things” in quotes.
• We call this the universal set. It՚s a set that contains everything. Well, not exactly everything. Everything that is relevant to our question.
• Two sets P and Q are equal if they have the same number of elements. A set P is a subset of a set Q if all the elements of P are also an element of Q.
• A power set [A (P) ] of a set P comprising of all subsets of P.
• The union of sets P and Q is a set comprising of all elements which are either in sets P or Q.
• Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pointless
• The intersection of sets P and Q is a set comprising of all common elements of sets P and Q.
• Similarly, the difference of sets P and Q in the same order is a set comprising of elements belonging to P but not Q.

## Important Equations

• For any two sets and ,
• If P and Q are finite sets such that P ∩ Q = φ, then n (P ∪ Q) = n (P) + n (Q) .
• If P ∩ Q ≠ φ, then
• If P is a subset of set U (Universal Set) , then its complement (P′) is also a subset of Universal Set (U) .

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