**Unformatted text preview: **( 2 ) The product of any fund irrational numbers, is irrational\
false - if *= fz and y : Vz then xy = ({ ` z ) = 2
2 is rational
15 ) a ) Prove that for all integers as if a is even then a is even
b ) Prove that $/Z is irrational.`
a ) Proof by contediction :`
Supose not , Suppose if as is even then a is aded.
Q = 2 1 + 1|
2 3 : ( 2 1C + 1 ) } - ( ZAC+ 1X 2/C+1 ) 2 FOIL`
- ( 2 1 6 + 1 ) ( 410 2 + 4 16 + 1 )|
gor`
- 8K + 8KC-12K+ 4 KC2 + 41C + 1 = 8KC # + 121 2 + 6K + 1^
= 2 ( 4 1 } + 6 K 2 + 3/10 ) + 1\
* 3 is odd by definition so this is a contactiction, therefore
the supposition is false and the theonion is True .\
6 ) Proof by contradiction
Suppose not , Suppose }/ Z is rational . Then there are integers a and & with no common for
So : 3/2 : 3 for some integers a and b and by of
= ( 3 / 2 ) 3 = (} ) }
- 2 : 23
= 9 3 : 25 3)
a ? is even by definition so a is even
a : EK for some integer IC
2 3 - 1 21) 3 = 8 K 3 = 25^ divide by 2 = 41 3- 5 3 4x 3- 2(ZK 3)
5 5 is even so b is even, but we know that a is even\
they share a common factor of 2 . This contradictsthe Supposition s`
the threrem is true ....

View
Full Document

- Fall '19
- NoProfessor
- Logic, Prime number, Rational number, 8KC-12K