**Proofs for De Morgan's laws :**

Here we are going to see the proof of De morgan's laws by Venn diagram.

**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)**

**De morgan's law for set complementation :**

Let U be the universal set containing sets A and B. Then

**(i) (A u B)' = A' n B'**

**(ii) (A n B)' = A' u B'**

**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.**

Now, let us look at the Venn diagram proof of De morgan's law for complementation.

**(A n B)' = A' u B'**

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

(A n B)' = A' u B'

**Hence, De morgan's law for complementation is verified.**

Let us look at some practice problems on "Demorgans law"

**Problem 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)**

**Similarly, one can verify ****A \ (B n C) = (A \ B) u (A \ C)**** for the given sets above.**

Let us look at the next problem on "Demorgans law"

**Problem 2 : **

Let U = { - 2, -1, 0, 1, 2, 3, 4, 5, .......10 }, A = {- 2, 2, 3, 4, 5 } and

B = { 1, 3, 5, 8, 9 }. Verify De Morgan’s laws of complementation

**Solution : **

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

To do this, we consider

A u B = {- 2, 2, 3, 4, 5 } u { 1, 3, 5, 8, 9 }

A u B = { -2, 1, 2, 3, 4, 5, 8, 9 }

We know that

(A u B)' = U \ { -2, 1, 2, 3, 4, 5, 8, 9 }

(A u B)' = { -1, 0, 6, 7, 10 } ---------(1)

A' = U \ A = U \ { -2, 2, 3, 4, 5 } = { -1, 0, 1, 6, 7, 8, 9, 10 }

B' = U \ B = U \ { 1, 3, 5, 8, 9 } = { -2, -1, 0, 2, 4, 6, 7, 10 }

A' n B' = { -1, 0, 1, 6, 7, 8, 9, 10 } n { -2, -1, 0, 2, 4, 6, 7, 10 }

A' n B' = { -1, 0, 6, 7, 10 } ---------(2)

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

**Similarly, one can verify (A n B)' = A' u B' for the given sets above.**

After having gone through the stuff given above, we hope that the students would have understood "Proofs for De Morgan's laws".

Apart from the stuff given above, if you want to know more about "Proofs for De Morgan's laws", please click here

Apart from "Demorgans law", if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**