The document discusses propositions, logical operations, predicates, quantification, and mathematical induction. It provides:
1) Definitions of predicates, propositions, universal and existential quantification, and the principle of mathematical induction.
2) Examples of applying predicates, quantification, and induction to prove statements about integers and sums.
3) The process of proving statements by mathematical induction, which involves showing the basis step and inductive step. It also introduces strong mathematical induction.
1. Propositions and Logical Operations
Definition: A predicate or a propositional function is a
noun/verb phrase template that describes a property of
objects, or a relationship among objects represented by
the variables:
Example: ๐๐ ๐ฅ๐ฅ : โ๐ฅ๐ฅ is integer less than 8.โ
๏ง ๐๐ 1 =
๏ง ๐๐ 10 =
๏ง ๐๐ โ11 =
1ยฉ S. Turaev, CSC 1700 Discrete Mathematics
2. Propositions and Logical Operations
Definition: The universal quantification of a predicate
๐๐ ๐ฅ๐ฅ is the statement โFor all values of ๐ฅ๐ฅ (for every ๐ฅ๐ฅ, for
each ๐ฅ๐ฅ, for any ๐ฅ๐ฅ), ๐๐ ๐ฅ๐ฅ is trueโ and is denoted by
โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ .
Example: ๐๐ ๐ฅ๐ฅ : โโ โ๐ฅ๐ฅ = ๐ฅ๐ฅโ is a predicate that is true
for all real numbers.
โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ =
Example: ๐๐ ๐ฅ๐ฅ : โ๐ฅ๐ฅ + 1 < 4โ.
โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ =
2ยฉ S. Turaev, CSC 1700 Discrete Mathematics
3. Propositions and Logical Operations
A predicate may contain several variables.
Example: ๐๐ ๐ฅ๐ฅ, ๐ฆ๐ฆ : ๐ฅ๐ฅ + ๐ฆ๐ฆ = ๐ฆ๐ฆ + ๐ฅ๐ฅ
โ๐ฅ๐ฅโ๐ฆ๐ฆ๐ฆ๐ฆ ๐ฅ๐ฅ, ๐ฆ๐ฆ =
Example: Write the following statement in the form of a
predicate and quantifier:
โThe sum of any two integers is even number.โ
3ยฉ S. Turaev, CSC 1700 Discrete Mathematics
4. Propositions and Logical Operations
Definition: The existential quantification of a predicate
๐๐ ๐ฅ๐ฅ is the statement โThere exists a value of ๐ฅ๐ฅ, for
which ๐๐ ๐ฅ๐ฅ is trueโ and is denoted by โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ .
Example: ๐๐ ๐ฅ๐ฅ : โโ๐ฅ๐ฅ = ๐ฅ๐ฅโ.
โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ =
Example: ๐๐ ๐ฅ๐ฅ : โ๐ฅ๐ฅ + 1 < 4โ.
โ๐ฅ๐ฅ๐ฅ๐ฅ ๐ฅ๐ฅ =
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6. Mathematical Induction
Suppose we have an infinite ladder:
1. We can reach the first rung of the
ladder.
2. If we can reach a particular rung of
the ladder, then we can reach the
next rung.
From (1), we can reach the first rung.
Then by applying (2), we can reach the
second rung. Applying (2) again, the
third rung, and so on. We can apply (2)
any # of times to reach any particular
rung, no matter how high up.
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7. Mathematical Induction
Principle of Mathematical Induction: To prove that ๐๐(๐๐)
is true for all positive integers ๐๐, we complete these
steps:
BASIS STEP: Show that ๐๐(๐๐0) is true.
INDUCTIVE STEP: Show that ๐๐ ๐๐ โ ๐๐(๐๐ + 1) is true
for all positive integers ๐๐ โฅ ๐๐0.
To complete the inductive step, assuming the inductive
hypothesis that ๐๐(๐๐) holds for an arbitrary integer ๐๐ โฅ
๐๐0, show that must ๐๐(๐๐ + 1) be true.
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8. Mathematical Induction
Example: Show that
๏ฟฝ
๐๐=1
๐๐
๐๐ =
๐๐(๐๐ + 1)
2
Solution:
BASIS STEP: ๐๐ 1 is true since
1 1 + 1
2
= 1.
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11. Mathematical Induction
Example: Prove that ๐๐ < 2๐๐ for all positive integers ๐๐.
Example: Prove that ๐๐! โฅ 2๐๐โ1
for all positive integers ๐๐.
Example: Prove that ๐๐3 โ ๐๐ is divisible by 3, for every
positive integer ๐๐.
Example: Let ๐ด๐ด1, ๐ด๐ด2, ๐ด๐ด3, โฆ , ๐ด๐ด๐๐ be any sets. Show that
๏ฟฝ
๐๐=1
๐๐
๐ด๐ด๐๐ = ๏ฟฝ
๐๐=1
๐๐
๐ด๐ด๐๐
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12. Strong Mathematical Induction
Strong induction tells us that we can
reach all rungs if:
1. We can reach the first rung of
the ladder.
2. For every integer ๐๐, if we can
reach the first ๐๐ rungs, then we
can reach the (๐๐ + 1)st rung.
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13. Strong Mathematical Induction
To conclude that we can reach every rung by strong
induction:
BASIS STEP: ๐๐ 1 holds.
INDUCTIVE STEP: Assume ๐๐ 1 โง ๐๐ 2 โง โฏ โง ๐๐ ๐๐
holds for an arbitrary integer ๐๐, and show that
๐๐ ๐๐ + 1 must also hold.
We will have then shown by strong induction that for
every positive integer ๐๐, ๐๐ ๐๐ holds, i.e., we can reach
the ๐๐th rung of the ladder.
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14. Strong Mathematical Induction
Example: Show that if ๐๐ is an integer greater than 1, then
๐๐ can be written as the product of primes.
Solution: Let ๐๐ ๐๐ be the proposition that ๐๐ can be
written as a product of primes.
BASIS STEP: ๐๐ 2 is true since 2 itself is prime.
INDUCTIVE STEP: The inductive hypothesis is ๐๐ ๐๐ is
true for all integers ๐๐ with 2 โค ๐๐ โค ๐๐. To show that
๐๐ ๐๐ + 1 must be true under this assumption, two
cases need to be considered:
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15. Strong Mathematical Induction
INDUCTIVE STEP: The inductive hypothesis is ๐๐ ๐๐ is
true for all integers ๐๐ with 2 โค ๐๐ โค ๐๐. To show that
๐๐ ๐๐ + 1 must be true under this assumption, two
cases need to be considered:
1. If ๐๐ + 1 is prime, then ๐๐ ๐๐ + 1 is true.
2. Otherwise, ๐๐ + 1 is composite and can be written
as the product of two positive integers ๐๐ and ๐๐
with 2 โค ๐๐ โค ๐๐ < ๐๐ + 1. By the inductive
hypothesis ๐๐ and ๐๐ can be written as the product
of primes and therefore ๐๐ + 1 can also be written
as the product of those primes.
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