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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Exercise 37 39 can be generalized as follows: If 0 k n and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly k elements is ( n k ) .

Use this result to write an expression for the total number of elements in the power set P ( A ) .

Use the binomial theorem as stated in Exercise 25 to evaluate the expression in part a and compare this result to exercise 27 and 37. (Hint: set a = b = 1 in the binomial theorem.)

If n is a nonnegative integer and the set A has n elements, then the power set P ( A ) has 2 n

If n 2 and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly two elements is ( n 2 ) = n ( n 1 ) 2 .

If n 3 and the set A has n elements, then the number of elements of the power set P ( A ) containing exactly three elements is ( n 3 ) = n ( n 1 ) ( n 2 ) 3 ! .

Let a and b be a real number, and let n be integers with 0 r n . The binomial theorem states that

( a + b ) n = ( n 0 ) a n + ( n 1 ) a n 1 b + ( n 2 ) a n 2 b 2 + ... + ( n r ) a n r b r + ....... + ( n n 2 ) a 2 b n 2 + ( n n 1 ) a b n 1 + ( n n ) b n

= r = 0 n ( n r ) a n r b r

Where the binomial coefficients ( n r ) are defined by

( n r ) = n ! ( n r ) ! r ! ,

With r ! = r ( r 1 ) ......... ( 2 ) ( 1 ) for r 1 and 0 ! = 1 . Prove that the binomial coefficients satisfy the equation

( n r 1 ) + ( n r ) = ( n + 1 r ) for 1 r n

(a)

To determine

An expression for the total number of elements in the power set P(A).

Explanation

Given information:

Result: If 0kn and the set A has n elements, then the number of elements of the power set P(A) containing exactly k elements is (nk).

Explanation:

It is given that if 0kn and the set A has n elements, then the number of elements of the power set P(A) containing exactly k elements is (nk).

The number of elements of the power set P(A) containing exactly 0 elements is (n0)

The number of elements of the power set P(A) containing exactly 1 elements is (n1

(b)

To determine

The value of expression in part a by using binomial theorem.

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