DE MORGANS LAW FOR SET DIFFERENCE

De morgans law for set difference :

Here we are going to see De morgan's law for set difference.

Demorgans law :

De Morgan’s father (a British national) was in the service of East India Company, India. Augustus De Morgan (1806-1871) was born in Madurai, Tamilnadu, India. His family moved to England when he was seven months old. He had his education at Trinity college, Cambridge, England.

De Morgan’s laws relate the three basic set operations union, intersection and complement.

De morgan's laws

De morgan's law for set difference :

For any three sets A, B and C, we have 

(i)  A \ (B u C)  =  (A \ B) n (A \ C)

(ii)  A \ (B n C)  =  (A \ B) u (A \ C)

Proof by Venn diagram

(i)  A \ (B n C)  =  (A \ B) u (A \ C)

From the above Venn diagrams (2) and (5), it is clear that 

A \ (B n C)  =  (A \ B) u (A \ C)

Hence, De morgan's law for set difference is verified.

(ii)  A \ (B u C)  =  (A \ B) n (A \ C)

Example 1 : 

Let A  =  { a, b, c, d, e, f, g, x, y, z }, B  =  { 1, 2, c, d, e } and C  =  { d, e, f, g, 2, y }. Verify De Morgan’s laws of set difference.

Solution : 

First, we shall verify A \ (B u C)  =  (A \ B) n (A \ C)

To do this, we consider

B u C  =  { 1, 2, c, d, e } u { d, e, f, g, 2, y }

B u C  =  { 1, 2, c, d, e, f, g, y }

We know that

A/(B u C) =  {a, b, c, d, e, f, g, x, y, z}\{1, 2, c, d, e, f, g, y}

A / (B u C)  =  { a, b, x, z } ---------(1)

A \ B   =  { a, b, f, g, x, y, z }

A \ C   =  { a, b, c, x, z }

(A \ B) n (A \ C)  =  { a, b, x, z } ---------(2)

From (1) and (2), it is clear that A\(B u C)  =  (A\B)n(A\C)

Example 2 :

Let A = {10,15, 20, 25, 30, 35, 40, 45, 50}, B = {1, 5,10,15, 20, 30} and C = {7, 8,15,20,35,45, 48}. Verify A \(B n C) = (A \ B) U (A \ C).

Solution :

First, we shall verify A \ (B u C)  =  (A \ B) n (A \ C)

To do this, we consider

B u C  = {1, 5, 10, 15, 20, 30} U {7, 8, 15, 20, 35, 45, 48}

B u C  =  { 1, 5, 7, 8, 10, 15 , 20,30, 35, 45, 48 }

We know that

A \ (B u C) =  {10,15, 20, 25, 30, 35, 40, 45, 50}\{ 1, 5, 7, 8, 10, 15 , 20,30, 35, 45, 48 }

A / (B u C)  =  { 25, 40, 50 } ---------(1)

A\B = {10, 15, 20, 25, 30, 35, 40, 45, 50}\{1, 5, 10, 15, 20, 30}

  =  {25, 35, 40, 45, 50}

A\C = {10, 15, 20, 25, 30, 35, 40, 45, 50}\ {7, 8, 15, 20, 35, 45, 48}

  =  {25, 30, 40, 50}

(A\B) U (A\C) = { 25, 40, 50} ---------(2)

From (1) and (2), it is clear that A\(B n C)  =  (A\B)u(A\C)

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  2. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More

  3. SAT Math Practice

    Mar 26, 24 08:53 PM

    satmathquestions1.png
    SAT Math Practice - Different Topics - Concept - Formulas - Example problems with step by step explanation

    Read More