Sets are defined as a well-defined collection of objects. 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. Two sets P and Q are equal if they have exactly 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. 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 P and Q,

- (P âˆª Q)â€² = Pâ€² âˆ© Qâ€²
- (P âˆ© Q)â€² = Pâ€² âˆª Qâ€²

- If P and Q are finite sets such that P âˆ© Q = Ï†, then n (P âˆª Q) = n (P) + n (Q).
- If P âˆ© Q â‰ Ï†, then

n (P âˆª Q) = n (P) + n (Q) â€“ n (P âˆ© Q)

- n (P âˆª Q âˆª R) = n(P) + n(Q) + n(R) â€“ n(P âˆ© Q) â€“ n(P âˆ© Q) â€“ n(P âˆ© Q ) + n(P âˆ© Q âˆ© R)
- If P is a subset of set U (Universal Set), then its complement (Pâ€²) is also a subset of Universal Set (U).

### Some Properties of Operation of Intersection

- P âˆ© Q = Q âˆ© P (Commutative law).
- (P âˆ© Q) âˆ© R = P âˆ© (Q âˆ© R) (Associative law).
- Ï† âˆ© P = Ï†, U âˆ© P = P.
- P âˆ© P = P (Idempotent law).
- P âˆ© (Q âˆª R) = (P âˆ© Q) âˆª (P âˆ© Q) (Distributive law).

### Some Properties of the Operation of Union

- P âˆª Q = Q âˆª P (Commutative law).
- (P âˆª Q) âˆª R = P âˆª ( Q âˆª R) (Associative law).
- P âˆª Ï† = P (Law of the identity element).
- U âˆª P = U (Law of U).

### Sets Class 11 Practice Questions

- For any set P and Q, Show that P = (P âˆ© Q) âˆª (P – Q) and P âˆª (Q â€“ P) = (P âˆª Q)
- Using the properties of sets, Prove

- P âˆª ( P âˆ© Q) = P
- P âˆ© (P âˆª Q) = P

- Prove that, P âˆ© Q = P âˆ© R need not imply Q = R.
- If P and Q are two sets and P âˆ© Z = Q âˆ© Z = Ï† and P âˆª Z = Q âˆª Z for some set Z, Prove that P = Q.
- Find sets M, N and O such that M âˆ© N, N âˆ© O, and M âˆ© O are non-empty sets and M âˆ© N âˆ© O = Ï†.
- Let M, N, and O are three sets such that M âˆª N = M âˆª O and M âˆ© N = M âˆ© O. Prove that N = O.
- If P âŠ‚ Q, then prove that R â€“ Q âŠ‚ Q â€“ P.
- If A(n) = A(m), then, prove that m = n.

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