Some rules of the sets in mathematics

In the set, we need to solve different types of problems. For that, we have to apply some rules of the sets. Previously we have discussed the Set in mathematics and its types.

Some rules of the sets

Some rules of the sets


Rules of the set union, intersection, and minus

A) If A, B, C are the subset of universal set U then-

  1. A∪A = A (A union B equals A)
  2. A∩A = A (A intersection B equals to A)
  3. A∪∅ = A (A union Empty set equals to A)
  4. A∩U= A (A intersection universal set equals to A)
  5. A∩∅ = ∅ (A intersection Empty set equals to Empty set)
Proves:
  1. In A∪A all elements are the union of set A and set A. Without repeating the same element we get set A∪A = A. 
  2. A∩A common elements of A and A will be the element of the set. So, A∩A =A.
  3. Because ∅ has no element so A∪∅=A.
  4. Because A is the subset of U so common elements of A and U will be the elements of A.
  5. Because ∅ has no elements so A and ∅ has no common elements.

B) If A, B are two sets then-

  1. A∪B = B∪A
  2. A∩B = B∩A
These two are the commutative laws of sets.

C) If A,B,C are 3 sets then-

  1. A∪(B∪C) = (A∪B)∪C
  2. A∩(B∩C) = (A∩B)∩C
These two are the associative laws of sets.

D) If A, B, C are thee sets then-
  1. A∪(B∪C) = (A∪B)∩(A∪C)
  2. A∩(B∩C) = (A∩B)∪(A∩C)
These laws are distributive laws of sets.

E)If A, B are the subset of universal set U then
  1. (A∪B)´ = Aˊ∩ Bˊ
  2. (A∩B)´ = A´∪ B´
This law is De Morgan's theorem.

So these were some rules of the sets. Now, read about the Ordered pairs and cartesian products in the set.

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