MODEL ANSWERS TO THE SECOND HOMEWORK 1. (a) σ is surjective. Indeed, let t ∈ R be a non-negative real number. Then t has a square root s ∈ R . In this case f ( s )= s 2 = t , by defnition oF the square root. σ is not injective. ±or example, σ (1) = σ (-1) = 1. (b) This is surjective, For the same reason as (a). It is now also injective, since every positive real number has a unique positive root. (c) Not surjective, since there is no positive integer whose square is two. Not injective, For the same reason as (a). (d) This is injective. Indeed suppose that f ( a )= f ( b ). Then 2 a =2 b , so that a = b . This is not surjective. Indeed there is not integer
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unique positive root, positive real number, non-negative real number