Use mathematical induction to prove that for every integer , if a set S has n elements, then the number of subsets of S with an even number of elements equals number of subsets of S with an odd number of elements.
To prove that the number of subsets of S with an even number of elements is equal to the number of subsets of S with an odd number of elements by mathematical induction.
For every integer , the set S has n elements.
The number of subsets for a set of elements is .
Let’s prove the result by mathematical induction.
The set S contains only one element.
The result is true for a 1-element set.
Assume that the result is true for k such that .
Let’s prove that this result is true for k+1-element set.
Let and .
That is set A contains k+1, elements and set B contains elements with the exception of the element of set B
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