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20190222_003843.jpg - 2 The product of any fund irrational...

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

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