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Quantifiers

Supplementary Notes

Prepared by Raymond WongPresented by Raymond Wong

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e.g.1 (Page 6)

We are going to prove the following claim C is true: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …

P(0) true

P(1) true

P(2) true

P(3) true

P(4) true

If we can prove that statement P(m) is true for each non-negative integer separately, then we can prove the above claim C is correct.

… true

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e.g.1

We are going to prove the following claim C is false: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …

P(0) true

P(1) true

P(2) true

P(3) true

P(4) true

… true

false

There may exist another non-negative integer k such that P(k) is false

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e.g.1

We are going to prove the following claim C is false: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …

P(0) false

P(1) false

P(2) true

P(3) true

P(4) true

… true

m2 > m integer 0, 1, 2, …-1, -2,…

02 > 0

12 > 1

22 > 2

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e.g.2 (Page 9)

We are going to prove the following claim C is true: there exists a non-negative integer m such that statement P(m) is true

P(0)

P(1)

P(2) true

P(3)

P(4)

If we can prove that statement P(m) is true for ONE non-negative integer, then we can prove the above claim C is correct.

…

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e.g.2

We are going to prove the following claim C is false: there exists a non-negative integer m such that statement P(m) is true

If we can prove that statement P(m) is false for each non-negative integer separately, then we can prove the above claim C is false.

P(0) false

P(1) false

P(2) false

P(3) false

P(4) false

… false

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e.g.2

We are going to prove the following claim C is true: there exists a non-negative integer m such that statement P(m) is true

true

22 > 2

P(0)

P(1)

P(2)

P(3)

P(4)

…

m2 > minteger

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e.g.3 (Page 13)

E.g. Using the quantifier notations, please re-write the Euclid’s division theorem that states

For every positive integer n and every non-negative integer m, there are integers q and r, with 0 r < n such that m = qn + r.

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e.g.3

For every positive integer n and every non-negative integer m, there are integers q and r, with 0 r < n such that m = qn + r.

Since m is non-negativeand n is a positive integer, we derive that q and r are also non-negative.

For every positive integer n and every non-negative integer m, there are non-negative integers q and r, with r < n such that m = qn + r.

Let Z+ be the set of positive integers.

Let N be the set of non-negative integers.

n Z+ ( )m N ( )q N ( )r N ( )(r < n) (m = qn + r)

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e.g.4 (Page 15)

n Z+ ( )m N ( )q N ( )r N ( )(r < n) (m = qn + r)

Let p(m, n, q, r) denote m = nq + r with r < n

n ( )m ( )q ( )r p(m, n, q, r)

If we remove the universe, then we can see the order in which the quantifieroccurs

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e.g.5 (Page 19) Is the following statement true?

x R+ (x > 1)If this statement is correct, we need to prove the following.

P(0) true

P(0.1) true

P(0.2) true

… true

P(1) true

… true

Let P(x) be “x > 1”

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e.g.5 Is the following statement true?

x R+ (x > 1)If this statement is incorrect, we need to prove the following.

false

P(0)

P(0.1)

P(0.2)

…

P(1)

…

Let P(x) be “x > 1”

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e.g.5 Is the following statement true?

x R+ (x > 1)If this statement is incorrect, we need to prove the following.

false

P(0)

P(0.1)

P(0.2)

…

P(1)

…

Let P(x) be “x > 1”

Consider x = 0.1

Note that 0.1 R+

“0.1 > 1” is false.

This statement is false.

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e.g.6 (Page 19) Is the following statement true?

x R+ (x > 1)If this statement is correct, we need to prove the following.

Let P(x) be “x > 1”

P(0)

P(0.1)

P(0.2) true

…

P(2)

…

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e.g.6 Is the following statement true?

x R+ (x > 1)If this statement is incorrect, we need to prove the following.

Let P(x) be “x > 1”

P(0) false

P(0.1) false

P(0.2) false

… false

P(2) false

… false

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e.g.6 Is the following statement true?

x R+ (x > 1)If this statement is correct, we need to prove the following.

Let P(x) be “x > 1”

P(0)

P(0.1)

P(0.2)

true

…

P(2)

…

Consider x = 2

Note that 2 R+

“2 > 1” is true.

This statement is true.

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e.g.7 (Page 19) Is the following statement true?

x R (y R (y > x))If this statement is correct, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2) true

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

true

x = 0

x = 0.1

x = 0.2 true

true

There exists a value y such that P(0, y) is true.

true

There exists a value y such that P(0.1, y) is true.

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e.g.7 (Page 19) Is the following statement true?

x R (y R (y > x))If this statement is incorrect, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2)

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

false

x = 0

x = 0.1

x = 0.2

There doest not exist a value y such that P(0.1, y) is true. That is, for each value y R,

P(0.1, y) is false.false

false

false

false

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e.g.7 Is the following statement true?

x R (y R (y > x))If this statement is correct, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2) true

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

true

x = 0

x = 0.1

x = 0.2 true

true

true

Let y = x + 1

Note that, if x R, then y R

“y > x” is true.

This statement is true.

y = 1

y = 1.1

y = 1.2

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e.g.8 (Page 19) Is the following statement true?

x R ( y R (y > x))If this statement is correct, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2) true

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

true

x = 0

x = 0.1

x = 0.2 true

true

true

true

true

true

true

true

true

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e.g.8 Is the following statement true?

x R ( y R (y > x))If this statement is incorrect, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2)

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

false

x = 0

x = 0.1

x = 0.2

false

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e.g.8 Is the following statement true?

x R ( y R (y > x))If this statement is incorrect, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0) P(0, 0.1)

P(0, 0.2)

…

P(0.1, 0) P(0.1, 0.1)

P(0.1, 0.2)

…

false

x = 0

x = 0.1

x = 0.2

false

Consider x = 0.1 and y = 0

Note that x R and y R

“y > x” is false.(i.e., “0 > 0.1” is false)

This statement is false.

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e.g.9 (Page 19) Is the following statement true?

x R ((x 0) y R+ (y > x))If this statement is correct, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0.1) P(0, 0.2)

P(0, 0.3) true

…

P(0.1, 0.1) P(0.1, 0.2)

P(0.1, 0.3)

…

x = 0

x = 0.1

x = 0.2

true

true

true

true

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e.g.9 Is the following statement true?

x R ((x 0) y R+ (y > x))If this statement is incorrect, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0.1) P(0, 0.2)

P(0, 0.3) false

…

P(0.1, 0.1) P(0.1, 0.2)

P(0.1, 0.3)

…

x = 0

x = 0.1

x = 0.2

false

false

false

false

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e.g.9 Is the following statement true?

x R ((x 0) y R+ (y > x))

x = 0.2

If this statement is correct, we need to prove the following.

Let P(x, y) be “ y > x”

P(0, 0.1) P(0, 0.2)

P(0, 0.3) true

…

P(0.1, 0.1) P(0.1, 0.2)

P(0.1, 0.3)

…

x = 0

x = 0.1

true

true

true

true

Let x = 0

Note that y R+

(i.e., y > 0)

“y > x” is true.(i.e., “y > 0” is true)

This statement is true.