Department

Computer ScienceCourse Code

CSC165H1Professor

Ilir DemaLecture

3This

**preview**shows page 1. to view the full**5 pages of the document.**CSC165H1 - Lecture 3 - Universally Quantified Implication

(Continued from Lecture 2)

Our solution is the following:

Let be a real number.

Assume +

√

❑

= 2.

Then

√

❑

= 2 −

x

.

Then

√

❑

¿

¿

¿

=

(2−x)❑2

.

Then =

(2−x)❑2

.

Then = 4 − + 4

x2

.

Then 0 = 4 − +5

x2

.

Then 0 = − 1) − 4). ( (

Then = 1 or = 4.

Therefore, for any number : if +

√

❑

= 2 then = 1 or = 4].[

We trace our solution using a tracing table.

If = 1 If = 4 Solution Set

+

√

❑

= 2 1+

√

❑

= 2? True 4 +

√

❑

= 2? False 1

√

❑

= 2 −

x

√

❑

= 2 − 1? True

√

❑

= 2 −

4

?

False 1

=

(2−x)❑2

1 =

(2−1)❑2

?True 4 =

(2−4)❑2

?True 1 or 4

= 4 − + 4

x2

1 = 4 − 4(1) +

12

?

True 4 = 4 − 4(4) +

42

?

True 1 or 4

0 = 4 − +5

x2

0 = 4 − 5(1) +

12

?

True 0 = 4 − 5(4) +

42

?

True 1 or 4

0 = − 1) − 4)( ( 0 = (1 − 1)(1 −

4)?

True 0 = (4 − 1)(4 −

4)?

True 1 or 4

= 1 or = 4 1 = 1 or 1 = 4? True 4 = 1 or 4 = 4? True 1 or 4

As we trace our solution, for it to be consistent the solution set has to stay the same or get bigger.

From our tracing table we can see that the solution gets bigger, which is consistent with the

consecutive pairs of lines representing true UQI’s and the “Then” statements used to connect them.

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Proof Process

+

√

❑

= 2? = 1 or = 4]?[

1 True True Okay

4 False True Irrelevant

0 False False Irrelevant

Using a number line and a Venn diagram…

A represents the set of all numbers such that +

√

❑

= 2

B represents the set of all numbers such that = 1 or = 4.

From the number line and the Venn diagram, we can see that A is completely contained in B but B is

not contained in A, because there are more elements in B than in A.

Understanding the Solution

Every step in the solution uses a known true Universally Quantified Implication. The first ‘Then’ part :

“Then

√

❑

= 2 −

x

.”, uses a known true UQI, specifically:

●For all numbers and and : if = then − = − c.

From this known one, we can deduce:

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