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.
The Set of Any Element is Termed an Wmpty 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:
There is a Simple Notation for Sets
  • 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) .

Some Properties of Operation of Intersection

  • .

Some Properties of the Operation of Union