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The point of this exercise is to get practice writing clear and complete solutions and to get feedback on your attempts. You will get participation credit for your posts and thoughtful comments.
4a. Prove that if x is a rational number and x ≠ 0 then 1/x is a rational number
If x is a rational number ≠ 0, then it can be written as a/b, where a and b are integers and ≠ 0. Then 1/x is equal to 1/(a/b) which equals b/a. Therefore if x is a rational number, 1/x is also rational.
4b. Prove that if x is an irrational number and x≠ 0 then 1/x is an irrational number.
Assume that x is irrational and 1/x is rational. Then 1/x can be written as a/b where a and b are integers ≠ 0. So x is equal to 1/(a/b) which equals b/a, b and a are integers. Therefore x is rational. This contradicts the assumption that x is irrational. Therefore it is true that if x is irrational, 1/x is also irrational.
3b. Use quantifiers and variables to translate the following propostion.
‘there is a tv show that everybody likes.
s= student
u= tv show
Ǝu∀s(T(s,u))