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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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For each of 5—21 prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set .

13. For all sets A, B, and C, A ( B C ) = ( A B ) ( A C ) .

To determine

To Prove:

For each prove each statement that is true and find a counterexample for each statement that is false. Assume all sets are subsets of a universal set .

For all sets A,B and C ,

A(BC)=(AB)(AC)

Explanation

Given information:

Let  A,B and C  be any set

Concept used:

:Union of sets:Intersection of sets

: subset of set

Calculation:

Suppose X and Y are two subsets of a universal set U. From the definition of differences and unions,

  • The difference Y minus X is denoted by YX and is defined as the set of all elements that are present in Y but not in X.
  • Also, the union X and Y is denoted by XY and is defined as the set of elements that are present in at least either X or Y.

Consider the statement,

A(BC)=(AB)(AC)

The above statement is false.

Consider a counter example where A={1,2,3},B={3,4,5} and C={2,4,6}.

Evaluate A(BC) corresponding to the counter example.

By the definition of difference.

BC={3,4,5}{2,4,6}={3,5}

Next find A(BC).

By the definition of union.

A(BC)={1,2,3}{3,5}={1,2,3,5}

Evaluate (AB)(AC) corresponding to the counter example

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Sect-6.1 P-11TYSect-6.1 P-1ESSect-6.1 P-2ESSect-6.1 P-3ESSect-6.1 P-4ESSect-6.1 P-5ESSect-6.1 P-6ESSect-6.1 P-7ESSect-6.1 P-8ESSect-6.1 P-9ESSect-6.1 P-10ESSect-6.1 P-11ESSect-6.1 P-12ESSect-6.1 P-13ESSect-6.1 P-14ESSect-6.1 P-15ESSect-6.1 P-16ESSect-6.1 P-17ESSect-6.1 P-18ESSect-6.1 P-19ESSect-6.1 P-20ESSect-6.1 P-21ESSect-6.1 P-22ESSect-6.1 P-23ESSect-6.1 P-24ESSect-6.1 P-25ESSect-6.1 P-26ESSect-6.1 P-27ESSect-6.1 P-28ESSect-6.1 P-29ESSect-6.1 P-30ESSect-6.1 P-31ESSect-6.1 P-32ESSect-6.1 P-33ESSect-6.1 P-34ESSect-6.1 P-35ESSect-6.1 P-36ESSect-6.1 P-37ESSect-6.1 P-38ESSect-6.2 P-1TYSect-6.2 P-2TYSect-6.2 P-3TYSect-6.2 P-4TYSect-6.2 P-5TYSect-6.2 P-6TYSect-6.2 P-1ESSect-6.2 P-2ESSect-6.2 P-3ESSect-6.2 P-4ESSect-6.2 P-5ESSect-6.2 P-6ESSect-6.2 P-7ESSect-6.2 P-8ESSect-6.2 P-9ESSect-6.2 P-10ESSect-6.2 P-11ESSect-6.2 P-12ESSect-6.2 P-13ESSect-6.2 P-14ESSect-6.2 P-15ESSect-6.2 P-16ESSect-6.2 P-17ESSect-6.2 P-18ESSect-6.2 P-19ESSect-6.2 P-20ESSect-6.2 P-21ESSect-6.2 P-22ESSect-6.2 P-23ESSect-6.2 P-24ESSect-6.2 P-25ESSect-6.2 P-26ESSect-6.2 P-27ESSect-6.2 P-28ESSect-6.2 P-29ESSect-6.2 P-30ESSect-6.2 P-31ESSect-6.2 P-32ESSect-6.2 P-33ESSect-6.2 P-34ESSect-6.2 P-35ESSect-6.2 P-36ESSect-6.2 P-37ESSect-6.2 P-38ESSect-6.2 P-39ESSect-6.2 P-40ESSect-6.2 P-41ESSect-6.2 P-42ESSect-6.2 P-43ESSect-6.2 P-44ESSect-6.3 P-1TYSect-6.3 P-2TYSect-6.3 P-3TYSect-6.3 P-1ESSect-6.3 P-2ESSect-6.3 P-3ESSect-6.3 P-4ESSect-6.3 P-5ESSect-6.3 P-6ESSect-6.3 P-7ESSect-6.3 P-8ESSect-6.3 P-9ESSect-6.3 P-10ESSect-6.3 P-11ESSect-6.3 P-12ESSect-6.3 P-13ESSect-6.3 P-14ESSect-6.3 P-15ESSect-6.3 P-16ESSect-6.3 P-17ESSect-6.3 P-18ESSect-6.3 P-19ESSect-6.3 P-20ESSect-6.3 P-21ESSect-6.3 P-22ESSect-6.3 P-23ESSect-6.3 P-24ESSect-6.3 P-25ESSect-6.3 P-26ESSect-6.3 P-27ESSect-6.3 P-28ESSect-6.3 P-29ESSect-6.3 P-30ESSect-6.3 P-31ESSect-6.3 P-32ESSect-6.3 P-33ESSect-6.3 P-34ESSect-6.3 P-35ESSect-6.3 P-36ESSect-6.3 P-37ESSect-6.3 P-38ESSect-6.3 P-39ESSect-6.3 P-40ESSect-6.3 P-41ESSect-6.3 P-42ESSect-6.3 P-43ESSect-6.3 P-44ESSect-6.3 P-45ESSect-6.3 P-46ESSect-6.3 P-47ESSect-6.3 P-48ESSect-6.3 P-49ESSect-6.3 P-50ESSect-6.3 P-51ESSect-6.3 P-52ESSect-6.3 P-53ESSect-6.3 P-54ESSect-6.4 P-1TYSect-6.4 P-2TYSect-6.4 P-3TYSect-6.4 P-1ESSect-6.4 P-2ESSect-6.4 P-3ESSect-6.4 P-4ESSect-6.4 P-5ESSect-6.4 P-6ESSect-6.4 P-7ESSect-6.4 P-8ESSect-6.4 P-9ESSect-6.4 P-10ESSect-6.4 P-11ESSect-6.4 P-12ESSect-6.4 P-13ESSect-6.4 P-14ESSect-6.4 P-15ESSect-6.4 P-16ESSect-6.4 P-17ESSect-6.4 P-18ESSect-6.4 P-19ESSect-6.4 P-20ESSect-6.4 P-21ESSect-6.4 P-22ESSect-6.4 P-23ESSect-6.4 P-24ESSect-6.4 P-25ESSect-6.4 P-26ESSect-6.4 P-27ESSect-6.4 P-28ESSect-6.4 P-29ES