The following properties of elements in sets are discussed here.

If U be the universal set and A, B and C are any three finite sets then;

**1.** If A and B are any two finite sets then n(A - B) = n(A) – n(A ∩ B) i.e. n(A – B) + n(A ∩ B) = n(A)

**2.** If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

**3.** If A and B are any two finite sets then n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint non-void sets.

**4.** If A and B are any two finite sets then n(A ∆ B) = Number of elements which belongs to exactly one of A or B

= n((A – B) ∪ (B – A))

= (A – B) + n(B – A) [Since (A - B) and (B – A) are disjoint.]

= n(A) – n(A ∩ B) + n(B) – n(A ∩ B)

= n(A) + n(B) – 2n(A ∩ B)

**Some more properties
of elements in sets using three finite sets**:

**5.**
If A, B and C are any three finite sets then n(A ∪ B ∪ C) = n(A)
+ n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A – C) + n(A ∩ B** **∩ C)

**6.**
If A, B and C are any three finite sets then Number of elements
in exactly one of the sets A, B, C = n(A) + n(B) + n(C) – 2n(A ∩ B) – 2n(B ∩ C)
– 2n(A – C) + 3n(A ∩ B** **∩ C)

**7.
**If A, B and C are any three finite sets then Number of elements
in exactly two of the sets A, B, C = n(A ∩ B) + n(B ∩ C) + n(C ∩ A) – 3n(A ∩ B ∩
C)

**8.** If U be the
universal set and A and B are any two finite sets then n(A' ∩
B') = n((A ∪ B)') = n(U) - n(A ∪ B)

**9.** If U be the
universal set and A and B are any two finite sets then n(A' ∪
B') = n((A ∩ B)') = n(U) - n(A ∩ B)

● **Set Theory**

● **Sets**

● **Subset**

● **Practice Test on Sets and Subsets**

● **Problems on Operation on Sets**

● **Practice Test on Operations on Sets**

● **Venn Diagrams in Different Situations**

● **Relationship in Sets using Venn Diagram**

● **Practice Test on Venn Diagrams**

**8th Grade Math Practice**

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