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# Points to Remember

A set is a well defined collection of objects. Sets are represented in three forms (i) Descriptive form (ii) Set ŌĆō builder form (iii) Roster form.

Set Language

Points to Remember

ŌĆó A set is a well defined collection of objects.

ŌĆó Sets are represented in three forms (i) Descriptive form (ii) Set ŌĆō builder form (iii) Roster form.

ŌĆó If every element of A is also an element of B, then A is called a subset of B.

ŌĆó If AŌŖåB and AŌēĀB, then A is a proper subset of B.

ŌĆó The power set of the set A is the set of all the subsets of A and it is denoted by P(A).

ŌĆó The number of subsets of a set with m elements is 2m.

ŌĆó The number of proper subsets of a set with m elements is 2m -1.

ŌĆó If AŌł®B = Ōłģ then A and B are disjoint sets. If AŌł®B ŌēĀ Ōłģ then A and B are overlapping.

ŌĆó The difference of two sets A and B is the set of all elements in A but not in B.

ŌĆó The symmetric difference of two sets A and B is the union of A-B and B-A.

### ŌĆó Commutative Property

For any two sets A and B,

AŌł¬B=BŌł¬A ; AŌł®B=BŌł®A

### ŌĆó Associative Property

For any three sets A, B and C

AŌł¬(B Ōł¬C)=(AŌł¬B)Ōł¬C ; AŌł®(B Ōł®C)=(AŌł®B)Ōł®C

### ŌĆó Distributive Property

For any three sets A, B and C

AŌł®(B Ōł¬C)=(AŌł®B)Ōł¬(AŌł®C)   Intersection over union

AŌł¬(B Ōł®C)=(AŌł¬B)Ōł®(AŌł¬C)   Union over intersection

### ŌĆó De MorganŌĆÖs Laws for Set Difference

For any three sets A, B and C

AŌłÆ(B Ōł¬C)=(AŌłÆB)Ōł®(AŌłÆC)

AŌłÆ(B Ōł®C)=(AŌłÆB)Ōł¬(AŌłÆC)

### ŌĆó De MorganŌĆÖs Laws for Complementation

Consider an Universal set and A, B are two subsets, then

(AŌł¬B)ŌĆ▓ =AŌĆ▓ Ōł® B ŌĆ▓ ; (AŌł®B)ŌĆ▓ =AŌĆ▓Ōł¬ BŌĆ▓

### ŌĆó Cardinality of Sets

If A and B are any two sets, then n (A Ōł¬ B ) = n(A) + n(B ) ŌłÆn(A Ōł® B)

If A, B and C are three sets, then

n (A Ōł¬ B Ōł¬C) = n (A) + n (B) + n (C) ŌłÆ n (A Ōł® B ) ŌłÆ n (B Ōł®C) ŌłÆ n (A Ōł®C ) + n (A Ōł® B Ōł®C)

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