Induction and real numbers
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A.prove by induction that (1^3+2^3+...+n^3)= [(1/2)n(n+1)]^2 for all n elements of N
b.give an example of two functions f,g on Reals to Reals such that f does not equal g but such that f o g = g o f.
c. show that if a,b elements of Real numbers then
i. max{a,b} = 1/2(a + b + abs(a-b)) and min{a,b}= 1/2(a+b-abs(a-b))
ii. min {a,b,c}= min{min{a,b},c}
d.Prove using math induction that if the sets S has n elements, then the power set of S, P(S), has 2^n elements.
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Solution Summary
This provides examples of several proofs using induction and working with functions and real numbers.
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Proof:
a.
When , is true. Suppose the equation is true, then for , we have
Thus the equation also holds for . Therefore, the equation holds for any positive integer .
b. Let and , then . But we have
Thus ...
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