Laws of Algebra of Sets (advance math)

Created with Sketch.

Laws of Algebra of Sets

 

Here we will learn about some of the laws of algebra of
sets.

1. Commutative Laws:

For any two finite sets A and B;

(i) A U B = B U A

(ii) A ∩ B = B ∩ A

 

2. Associative Laws:

For any three finite sets A, B and C;

(i) (A U B) U C = A U (B U C)

(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

Thus, union and intersection are associative.

 

3. Idempotent Laws:

For any finite set A;

(i) A U A = A

(ii) A ∩ A = A

 

4. Distributive Laws:

For any three finite
sets A, B and C;

(i) A U (B ∩ C) = (A U
B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩
B) U (A ∩ C)

Thus, union and intersection are distributive over
intersection and union respectively.

 

5. De Morgan’s Laws:

For any two finite
sets A and B;

(i) A – (B U C) = (A – B) ∩
(A – C)

(ii) A – (B ∩
C) = (A – B) U (A – C)

De Morgan’s Laws can also we written as:

(i) (A U B)’ =
A’ ∩ B’

(ii) (A ∩
B)’ = A’ U B’

 

More laws of algebra
of sets:

6. For any two
finite sets A and B;

(i) A – B = A ∩
B’

(ii) B – A = B ∩ A’

(iii) A – B = A ⇔ A ∩ B = ∅

(iv) (A – B) U B = A U B

(v) (A – B) ∩
B = ∅

(vi) A ⊆ B ⇔ B’ ⊆ A’

(vii) (A – B) U (B – A) = (A U B) – (A ∩ B)

 

7. For any three finite sets A, B and C;

(i) A – (B ∩ C) = (A –
B) U (A – C)

(ii) A – (B U C) = (A –
B) ∩ (A – C)

(iii) A ∩ (B – C) = (A ∩
B) – (A ∩ C)

(iv) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)

 

Leave a Reply

Your email address will not be published. Required fields are marked *

This is a free online math calculator together with a variety of other free math calculatorsMaths calculators
+