Consider the proposition ∀x ∈ R, if x 2 is irrational then x is irrational. Write down the contrapositive and try to prove it (you may take it as given that the product of two integers is also an integer)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Consider the proposition ∀x ∈ R, if x 2 is irrational then x is irrational. Write down
the contrapositive and try to prove it (you may take it as given that the product of
two integers is also an integer).

Expert Solution
Step 1

The given statement is xR, if x2 is irrational then x is irrational. Prove using the contrapositive method. To prove the given statement, prove that xR, if x is not irrational then x2 is not irrational.

Consider x is not irrational and x2 is irrational.

Therefore x is rational, therefore, x=pqwhere gcdp,q=1 and q0.

 

 

 

 

 

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