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Formulas for Set Theory

March 7, 2024

Set theory and its Formula

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. Formulas for Set Theory are very useful in terms of exams.

Formulas for Set Theory

Formulas for Set Theory

Notations used in set theory:

  • n(A) – Cardinal number of set A.
  • n∘(A) – Cardinality of set A.
  • \overline{A} = Ac – complement of set A.
  • U – Universal
  • A ⊂ B – Set A is proper subset of subset of B.
  • A ⊆ B – Set A is subset of set B.
  • ∅ – Null set.
  • a ∈ A – element “a” belongs to set A.
  • A ∪ B – union of set A and set B.
  • A ∩ B – intersection of set A and set B.

Formulas

  1. If A and B are overlapping set, n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
  2. If A and B are disjoint set, n(A ∪ B) = n(A) + n(B).
  3. n(A) = n(A ∪ B) + n(A ∩ B) – n(B).
  4. n(A ∩ B) = n(A) + n(B) – n(A ∪  B).
  5. n(B) = n(A ∪ B) + n(A ∩ B) – n(A).
  6. n(U) = n(A) + n(B) – n(A ∩ B) + n((A ∪ B)c).
  7. n((A ∪ B)c) = n(U) + n(A ∩ B) – n(A) – n(B).
  8. n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B).
  9. n(A – B) = n(A ∪  B) – n(B).
  10. n(A – B) = n(A) – n(A ∩ B).
  11. n(Ac) = n(U)- n(A)

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Some Examples based On Above Formulas:

Question 1: If A = { 1, 4, 5, 8} and B = { 6, 8, 10, 12}. Find A ∩ B.

Answer: A∩B= {8}

set theory formulas

Question 2: Check whether A={8,7,6,9} and B={9,6,7,8} are equal sets?

Answer: Yes they are equal sets since all the elements of A are present In B.

Set theory formula

Question 3: If A= { 7,8,0,6,8} B = { 5,6,7,8,9,0} Find A-B and B-A?

Answer: A-B= Elements in set a but not in b = { } null:

B-A= Elements in set b but not in a= {5, 9}.

set theory formulas

Question 4:If A={1,2,3,4} B={3,5,6,9} and C={6,7,8,9} Find A∩(B ∪ C).

Answer: B ∪ C = {6,9}, A∩(B ∪ C) = { } null, as there is no element in common

Set theory formulas

Question 5: If U={ 1,2,3,4,5,6,7,8,9} A= {2,4,6} Find A’.

Answer: A’ means all the elements except that in A.   {1,3,5,7,8,9}

Set theory formulas

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