Find the mistake in the following “proof” that purports to show that every nonnegative integer power of every nonzero real number is 1. “Proof: Let r be any nonzero real number and let the property P ( n ) be the equation r n = 1 . Show that P (0) is true: P (O) is true because r 0 = 1 by definition of zeroth power. Show that for every integer k ≥ 0 , if P(i) is True far each integer i from 0 through k, then P ( k + 1 ) is also true: Let k be any integer with k ≥ 0 and suppose that r i = 1 for each integer i from 0 through k . This is the inductive hypothesis We must show that r k + 1 = 1 . Now r k + 1 = r k + k − ( k − 1 ) because k + k − ( k − 1 ) = k + k − k + 1 = r k ⋅ r k r k − 1 by the laws of exponents = 1 ⋅ 1 1 by Inductive hypothesis = 1. Thus r k + 1 = 1 [as was to be shown]. [Since we have: proved the basis step mm! the inductive step of the strong mathematical induction, we conclude that the given statement is true.]”
Find the mistake in the following “proof” that purports to show that every nonnegative integer power of every nonzero real number is 1. “Proof: Let r be any nonzero real number and let the property P ( n ) be the equation r n = 1 . Show that P (0) is true: P (O) is true because r 0 = 1 by definition of zeroth power. Show that for every integer k ≥ 0 , if P(i) is True far each integer i from 0 through k, then P ( k + 1 ) is also true: Let k be any integer with k ≥ 0 and suppose that r i = 1 for each integer i from 0 through k . This is the inductive hypothesis We must show that r k + 1 = 1 . Now r k + 1 = r k + k − ( k − 1 ) because k + k − ( k − 1 ) = k + k − k + 1 = r k ⋅ r k r k − 1 by the laws of exponents = 1 ⋅ 1 1 by Inductive hypothesis = 1. Thus r k + 1 = 1 [as was to be shown]. [Since we have: proved the basis step mm! the inductive step of the strong mathematical induction, we conclude that the given statement is true.]”
Solution Summary: The author explains the mistake in the proof of P(k+1).
Find the mistake in the following “proof” that purports to show that every nonnegative integer power of every nonzero real number is 1.
“Proof: Let r be any nonzero real number and let the property P(n) be the equation
r
n
=
1
.
Show that P(0) is true: P(O) is true because
r
0
=
1
by definition of zeroth power.
Show that for every integer
k
≥
0
, if P(i) is True far each integer i from 0 through k, then
P
(
k
+
1
)
is also true: Let k be any integer with
k
≥
0
and suppose that
r
i
=
1
for each integer i from 0 through k. This is the inductive hypothesis We must show that
r
k
+
1
=
1
. Now
r
k
+
1
=
r
k
+
k
−
(
k
−
1
)
because
k
+
k
−
(
k
−
1
)
=
k
+
k
−
k
+
1
=
r
k
⋅
r
k
r
k
−
1
by the laws of exponents
=
1
⋅
1
1
by Inductive hypothesis
=
1.
Thus
r
k
+
1
=
1
[as was to be shown].
[Since we have: proved the basis step mm! the inductive step of the strong mathematical induction, we conclude that the given statement is true.]”
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MFCS unit-1 || Part:1 || JNTU || Well formed formula || propositional calculus || truth tables; Author: Learn with Smily;https://www.youtube.com/watch?v=XV15Q4mCcHc;License: Standard YouTube License, CC-BY