Science:Math Exam Resources/Courses/MATH220/December 2011/Question 04
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Question 04 

Let ƒ: A → B be a function. If C ⊆ A, define Prove that if ƒ is injective, then 
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Hint 

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We want to show that and that Let us start with the first. Suppose that ; that is, there is some such that . Since and , it follows that and that , i.e. that as desired. Note: This proof in no way uses injectivity; this is a general statement about all functions. Now we prove the second statement. Suppose that . That is, and . Equivalently, there is some such that , and some such that . We claim that , and hence . But this follows immediately from the injectivity of . It follows now, since and that as claimed. 