Distributive Law Definition

The distributive law is placed in set theory. The distributive law is the piece of sets that used to learn about the sets. Group of items is called the sets. Set theory have some operations union, intersection, difference, complement and performs some law.The set theory is used to select the number of objects simultaneously. Here we will see about the distributive law definition with examples.

Examples - Distributive Law Definition:

Now we are going to solve the examples for distributive law definition.

AU (B∩C) = (AUB) ∩ (AUC)

A∩ (BUC) = (A∩B) U (A∩C)

Example 1

Get the solution of the given sets by using distributive law. A={ a,b,c,d } B={ b,c,g } and C={ b,c,k,m }.

Solution:

The given sets are A={a,b,c,d} B={b,c,g} and C={b,c,k,m}.

Distributive law for set is as follows,

i)AU (B∩C)=(AUB) ∩(AUC)

ii)A∩(BUC)=(A∩B) U (A∩C)

i)AU (B∩C)=(AUB) ∩(AUC)

First find the solution left hand side.

AU (B∩C)

First we can solve the inner bracket set.

B∩C

Intersection means we choose the common elements from the set B and C.

So B∩C =   B= {b, c, g} ∩ C= {b, c, k, m}.

B∩C = {b, c}

Now find the AU (B∩C)

Grouping the values of A and B∩C

AU (B∩C) = {a, b, c, d} U (b, c)

= {a, b, c, d}

So AU (B∩C) = {a, b, c, d} -------------- (1)

(AUB) ∩ (AUC)

Groping the values of A and B sets.

A= {a, b, c, d} B= {b, c, g}

AUB = A= {a, b, c, d} U B= {b, c, g}

= {a, b, c, d, g}

Joining the values of A and C sets.

AUC:

A= {a, b, c, d} U C= {b, c, k, m}.

= {a, b, c, d, k, m}

(AUB) ∩ (AUC):

AUB ∩AUC = {a, b, c, d, g} ∩ {a, b, c, d, k, m}

(AUB) ∩ (AUC) = {a, b, c, d} ------------- (2)

So AU (B∩C) = (AUB) ∩ (AUC)

ii) A∩ (BUC) = (A∩B) U (A∩C)

A={ a,b,c,d } B={ b,c,g } and C={ b,c,k,m }.

Now solve the left side.

A∩ (BUC)

BUC:

= {b, c, g} U {b, c, k, m}

BUC = {b, c, g, k, m}

A∩ (B U C):

A= {a, b, c, d} ∩ {b, c, g, k, m}

= {b, c} -------------- (1)

(A∩B) U (A∩C)

A∩B= {a, b, c, d} ∩ B= {b, c, g}

= {b, c}

A∩C= A={ a,b,c,d } ∩ C={ b,c,k,m }.

= {b, c}

(A∩B) U (A∩C) = {b, c}

A∩ (BUC) = (A∩B) U (A∩C)


Example 2 – Distributive Law Definition:

A= {2, 7, 9} B= {5, 10, 11} C= {5, 8, 10}

What is AU (B∩C)

(B∩C):

= {5, 10, 11} ∩ {5, 8, 10}

= {5, 10}

AU (B∩C):

= {2, 7, 9} U {5, 10}

= {2, 7, 9, 5, 10}

These are examples for distributive law definition.

That’s all about distributive law definition.