**Properties of set operations :**

When two or more sets are combined together to form another set under some given conditions, then operations on sets are carried out.

The following are the important properties of set operations.

**(i) COMMUTATIVE PROPERTY **

(a) A u B = B u A (Set union is commutative)

(b) A n B = B n A (Set intersection is commutative)

**(ii) ASSOCIATIVE PROPERTY**

(a) A u (B u C) = (A u B) u C

(Set union is associative)

(b) A n (B n C) = (A n B) n C

(Set intersection is associative)

**(iii) DISTRIBUTIVE PROPERTY**

(a) A n (B u C) = (A n B) u (A n C)

(Intersection distributes over union)

(a) A u (B n C) = (A u B) n (A u C)

(Union distributes over intersection)

**Problem 1 :**

For the given sets A = { -10, 0, 1, 9, 2, 4, 5 } and B = {-1, -2, 5, 6, 2, 3, 4 }, verify that

(i) Set union is commutative. Also verify it by using Venn diagram.

(ii) Set intersection is commutative. Also verify it by using Venn diagram.

**Solution : **

(i) Let us verify that union is commutative.

A u B = { -10, 0, 1, 9, 2, 4, 5 } u {-1, -2, 5, 6, 2, 3, 4 }

A u B = { -10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9 } ---------(1)

B u A = {-1, -2, 5, 6, 2, 3, 4 } u { -10, 0, 1, 9, 2, 4, 5 }

B u A = { -10, -2, -1, 0, 1, 2, 3, 4, 5, 6, 9 } ---------(2)

From (1) and (2), we have

A u B = B u A

By Venn diagram, we have

From the above two Venn diagrams, it is clear that

A u B = B u A

**Hence, it is verified that set union is commutative.**

(ii) Let us verify that union is commutative.

A n B = { -10, 0, 1, 9, 2, 4, 5 } n {-1, -2, 5, 6, 2, 3, 4 }

A n B = { 2, 4, 5 } ---------(1)

B n A = {-1, -2, 5, 6, 2, 3, 4 } u { -10, 0, 1, 9, 2, 4, 5 }

B n A = { 2, 4, 5 } ---------(2)

From (1) and (2), we have

A n B = B n A

By Venn diagram, we have

From the above two Venn diagrams, it is clear that

A n B = B n A

**Hence, it is verified that set intersection is commutative.**

Let us look at the next problem on "Properties of set operations"

**Problem 2 :**

For the given sets A = { 1, 2, 3, 4, 5 }, B = { 3, 4, 5, 6 } and C = { 5, 6, 7, 8 }, verify that A u (B u C ) = (A u B) u C. Also verify it by using Venn diagram.

**Solution : **

Let us verify that set union is associative.

B u C = { 3, 4, 5, 6 } u { 5, 6, 7, 8 }

B u C = { 3, 4, 5, 6, 7, 8 }

A u (B u C) = { 1, 2, 3, 4, 5 } u { 3, 4, 5, 6, 7, 8 }

A u (B u C) = { 1, 2, 3, 4, 5, 6, 7, 8 } ---------(1)

A u B = { 1, 2, 3, 4, 5 } u { 3, 4, 5, 6 }

A u B = { 1, 2, 3, 4, 5, 6 }

(A u B) u C = { 1, 2, 3, 4, 5, 6 } u { 5, 6, 7, 8 }

(A u B) u C = { 1, 2, 3, 4, 5, 6, 7, 8 } ---------(2)

From (1) and (2), we have

A u (B u C) = (A u B) u C

By Venn diagram, we have

From the above Venn diagrams (2) and (4), it is clear that

A u (B u C) = (A u B) u C

**Hence, it is verified that set union is associative.**

Let us look at the next problem on "Properties of set operations"

**Problem 3 :**

For the given sets A = { a, b, c, d }, B = { a, c, e } and C = { a, e }, verify that A n (B n C) = (A n B) n C. Also verify it by using Venn diagram.

**Solution : **

Let us verify that set intersection is associative.

B n C = { a, c, e } u { a, e }

B n C = { a, e }

A n (B n C) = { a, b, c, d } n { a, e }

A n (B n C) = { a } ---------(1)

A n B = { a, b, c, d } u { a, c, e }

A n B = { a, c }

(A n B) n C = { a, c } n { a, e }

(A n B) n C = { a } ---------(2)

From (1) and (2), we have

A n (B n C) = (A n B) n C

By Venn diagram, we have

From the above Venn diagrams (2) and (4), it is clear that

A n (B n C) = (A n B) n C

**Hence, it is verified that set intersection is associative.**

Let us look at the next problem on "Properties of set operations"

**Problem 4 :**

For the given sets A = { 0, 1, 2, 3, 4 }, B = { 1, -2, 3, 4, 5, 6 } and C = { 2, 4, 6, 7 }, verify that A u (B n C ) = (A u B) n (A u C). Also verify it by using Venn diagram.

**Solution : **

Let us verify that union distributes over intersection.

B n C = { 1, -2, 3, 4, 5, 6 } n { 2, 4, 6, 7 }

B n C = { 4, 6 }

A u (B n C) = { 0, 1, 2, 3, 4 } u { 4, 6 }

A u (B n C) = { 0, 1, 2, 3, 4, 6 } ---------(1)

A u B = { 0, 1, 2, 3, 4 } u { 1, -2, 3, 4, 5, 6 }

A u B = { -2, 0, 1, 2, 3, 4, 5, 6 }

A u C = { 0, 1, 2, 3, 4 } u { 2, 4, 6, 7 }

A u C = { 0, 1, 2, 3, 4, 6, 7 }

(A u B) n (A u C) = { -2, 0, 1, 2, 3, 4, 5, 6 } n { 0, 1, 2, 3, 4, 6, 7 }

(A u B) n (A u C) = { 0, 1, 2, 3, 4, 6 } ---------(1)

From (1) and (2), we have

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

By Venn diagram, we have

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

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

**Hence, it is verified that union distributes over intersection.**

After having gone through the stuff given above, we hope that the students would have understood "Properties of set operations".

Apart from the stuff given above, if you want to know more about "Properties of set operations", please click here

Apart from "Properties of set operations", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM 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**