mathematical induction for sets
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Mathematical Induction for Sets
Let S be a subset of the positive integers.
If
(i) a 2 S, and
(ii) k 2 S ! k + 1 2 S for any arbitrary k 2 Z+,
then S contains every integer greater than or equal
to a.
Mathematical Induction for Sets
Let S be a subset of the positive integers.
If
(i) a 2 S, and
(ii) k 2 S ! k + 1 2 S for any arbitrary k 2 Z+,
then S contains every integer greater than or equal
to a.
Mathematical Induction for Sets
Let S be a subset of the positive integers.
If
(i) a 2 S, and
(ii) k 2 S ! k + 1 2 S for any arbitrary k 2 Z+,
then S contains every integer greater than or equal
to a.
Mathematical Induction for Sets
Let S be a subset of the positive integers.
If
(i) a 2 S, and
(ii) k 2 S ! k + 1 2 S for any arbitrary k 2 Z+,
then S contains every integer greater than or equal
to a.
Mathematical Induction for Sets
Let S be a subset of the positive integers.
If
(i) a 2 S, and
(ii) k 2 S ! k + 1 2 S for any arbitrary k 2 Z+,
then S contains every integer greater than or equal
to a.
Example. Use mathematical induction to prove
that
1 + 2 + 3 + . . .+ n =n(n+ 1)
2for all positive integers.
Mathematical Induction for Predicates
Let P (n) be a sentence whose domain is the
positive integers.
Suppose that:
(i) P (a)
(ii) P (k) ! P (k + 1) for any arbitrary k 2 Z+
Then P (n) is true for all positive integers greater
than or equal to a.
Mathematical Induction for Predicates
Let P (n) be a sentence whose domain is the
positive integers.
Suppose that:
(i) P (a)
(ii) P (k) ! P (k + 1) for any arbitrary k 2 Z+
Then P (n) is true for all positive integers greater
than or equal to a.
Mathematical Induction for Predicates
Let P (n) be a sentence whose domain is the
positive integers.
Suppose that:
(i) P (a)
(ii) P (k) ! P (k + 1) for any arbitrary k 2 Z+
Then P (n) is true for all positive integers greater
than or equal to a.
Mathematical Induction for Predicates
Let P (n) be a sentence whose domain is the
positive integers.
Suppose that:
(i) P (a)
(ii) P (k) ! P (k + 1) for any arbitrary k 2 Z+
Then P (n) is true for all positive integers greater
than or equal to a.
Mathematical Induction for Predicates
Let P (n) be a sentence whose domain is the
positive integers.
Suppose that:
(i) P (a)
(ii) P (k) ! P (k + 1) for any arbitrary k 2 Z+
Then P (n) is true for all positive integers greater
than or equal to a.
Example. Use mathematical induction to prove
that there are infinitely many prime numbers.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Mathematical Induction for Dummies
Blah blah some statement involving n blah blah.
(i) BASE CASE: Show true for n = a.
(ii) INDUCTIVE HYPOTHESIS: Assume true
for n = k (for some arbitrary k 2 Z+).
(iii) INDUCTIVE STEP: Show true for
n = k + 1 using inductive hypothesis.
Then we’ve proved it’s true for all positive integers
greater than or equal to a.
Example. Use mathematical induction to establish
the truth of the following statement for n � 0:
nX
i=0
2i = 2n+1 � 1
Example. Use mathematical induction to establish
the truth of the following statement for n � 0:
nX
i=0
2i = 2n+1 � 1
Example. Use mathematical induction to show:
nX
i=1
i3 =n2(n+ 1)2
4
for all positive integers.
