Identifying a Mistake in a Proposed Proof Find the mistake in the following "proof." Theorem: If n is any even integer, then (-1)" = 1. Proof: 1. Suppose n is any even integer. [We must show that (-1)" = 1.] 2. By definition of even, n = 2a for some integer a. 3. Then (-1)" = (-1)2a alldw = (-1)*)² ri by substitution 4. (-1)")? r by a law of exponents %3D 5. because any nonzero real number squared is positive.
Identifying a Mistake in a Proposed Proof Find the mistake in the following "proof." Theorem: If n is any even integer, then (-1)" = 1. Proof: 1. Suppose n is any even integer. [We must show that (-1)" = 1.] 2. By definition of even, n = 2a for some integer a. 3. Then (-1)" = (-1)2a alldw = (-1)*)² ri by substitution 4. (-1)")? r by a law of exponents %3D 5. because any nonzero real number squared is positive.
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Can someone explain the solution to this question in simplest terms in 4.2.2?????????
![you to identify mistakes in “proofs" that have been proposed. Example 4.2.2 illustrates the
kind of care that must be taken in evaluating a proof.
e, but to mak
ion for writing
s odd. Use o
not any othe
Example 4.2.2
Identifying a Mistake in a Proposed Proof
Find the mistake in the following "proof."
Theorem: If n is any even integer, then (-1)" = 1.
Proof:
1. Suppose n is any even integer. [We must show that (–1)" = 1.]
2. By definition of even, n =
2a for some integer a.
nent using na
3. Then
(-1)" = (-1)2ª
crie by substitution
ient way to rek
Idw = ((-1))² lo by a law of exponents
d od qo l asir
do llul isd 4.
5 is odd.
Dad
= 1
5.
because any nonzero real number squared is positive.
%3D
TU GLEOR
nce a - bisa
banland
nolyeT bu
Solution
This "proof" incorrectly jumps to a conclusion in line 5. Although it is true that the square
of any nonzero real number is positive, it does not follow that the square of (-1)ª is 1.
Exercise 10 at the end of this section asks you to give a correct proof of this theorem.
is odd.
Suppose a sa
d be "We mus
d and even,yu
al, section ini
nd long boarvs
Showing That an Existential Statement Is False
Recall that the negation of an existential statement is universal. It follows that to prove an
existential statement is false, you must prove a universal statement (its negation) is true.
Example 4.2.3
Disproving an Existential Statement
Show that the following statement is false:
There is a positive integer n such that n+ 3n+ 2 is prime.
Solution Proving that the given statement is false is equivalent to proving its negation is
true. The negation is
must show
For all positive integers n, n + 3n +2 is not prime.
Because the negation is universal, it is proved by generalizing from the generic particular.
Claim: The statement "There is a positive integer n such that n + 3n +2 is prime" is false.
2s for some](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7daa69d8-205a-4b72-abf5-96966d755823%2F868e09e2-8424-4f9b-9e03-a1b9583d1f3c%2Fz5nvbq7b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:you to identify mistakes in “proofs" that have been proposed. Example 4.2.2 illustrates the
kind of care that must be taken in evaluating a proof.
e, but to mak
ion for writing
s odd. Use o
not any othe
Example 4.2.2
Identifying a Mistake in a Proposed Proof
Find the mistake in the following "proof."
Theorem: If n is any even integer, then (-1)" = 1.
Proof:
1. Suppose n is any even integer. [We must show that (–1)" = 1.]
2. By definition of even, n =
2a for some integer a.
nent using na
3. Then
(-1)" = (-1)2ª
crie by substitution
ient way to rek
Idw = ((-1))² lo by a law of exponents
d od qo l asir
do llul isd 4.
5 is odd.
Dad
= 1
5.
because any nonzero real number squared is positive.
%3D
TU GLEOR
nce a - bisa
banland
nolyeT bu
Solution
This "proof" incorrectly jumps to a conclusion in line 5. Although it is true that the square
of any nonzero real number is positive, it does not follow that the square of (-1)ª is 1.
Exercise 10 at the end of this section asks you to give a correct proof of this theorem.
is odd.
Suppose a sa
d be "We mus
d and even,yu
al, section ini
nd long boarvs
Showing That an Existential Statement Is False
Recall that the negation of an existential statement is universal. It follows that to prove an
existential statement is false, you must prove a universal statement (its negation) is true.
Example 4.2.3
Disproving an Existential Statement
Show that the following statement is false:
There is a positive integer n such that n+ 3n+ 2 is prime.
Solution Proving that the given statement is false is equivalent to proving its negation is
true. The negation is
must show
For all positive integers n, n + 3n +2 is not prime.
Because the negation is universal, it is proved by generalizing from the generic particular.
Claim: The statement "There is a positive integer n such that n + 3n +2 is prime" is false.
2s for some
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