Unformatted text preview: COT 3100 Discrete Mathematics  Quiz #2
() 1. Use rule of inference to show that if
(
true, then ( ( )
( )) is also true.
Step
( 2. () ( ))
() 3. () 4. (() 5.
6.
7.
8.
9.
10. 12. () () ( )) and (() ( )) () (()
() ( )) Implication using (5) Premise
Universal instantiation using (8) ()
() (() Premise Resolution using (3) and (6) () () Universal instantiation using (1) Universal instantiation using (4) () () Premise Implication using (2) () () 11. ( ) (() Reason () 1. ( )) Implication using (9)
Resolution using (7) and (10) ( )) Universal generalization using (11) 2. Given x + y is irrational and x + z is rational, prove or disprove the following statements:
a) y  z is irrational.
b) y + z is irrational.
a) Proof: Suppose yz is rational. Then yz for some integer a and b. Since x+z is rational, x+z= for some integer c and d.
Thus x+y=(yz)+(x+z)= . Hence x+y is rational contradicting the given fact.
b) y+z can be rational or irrational.
Rational example: x=√ , y=√ , z= √ .
Then x+y= √ , x + z=0, y+z=0. ( )) are Irrational example:
x=√ , y= √ , z= √ .
Then x+y= √ , x + z=0, y+z=√ .
Therefore, y + z can be either rational or irrational
3. If
(
the statement. ) is true, does it necessarily follow that This statement is false. Because there may not be a letting all
For example P(x,y,z)={ , for real number x, y, z}. ( works for ( ) is true? Prove or disprove ) ...
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 Spring '08
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 Logic, universal instantiation

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