Answers to Some of the Homework Problems
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چکیده
1. We just write out (i). Recall that X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2}. Hence x ∈ X1 =⇒ x ∈ X1 ∪ X2, so that X1 ⊆ X1 ∪ X2. By symmetry, X1 ⊆ X1 ∪ X2. Finally, suppose that Y is a set such that X1 ⊆ Y and X2 ⊆ Y . Let x ∈ X1 ∪X2. Then x ∈ X1 or x ∈ X2. If x ∈ X1, then x ∈ Y since X1 ⊆ Y . Similarly, if x ∈ X2, then x ∈ Y . Thus in either case x ∈ Y , so that X1 ∪X2 ⊆ Y . 2. (i) Injective, not surjective, not bijective; image = (0,∞). (ii) Injective and surjective, hence bijective; image = R. (iii) Injective and surjective, hence bijective; image = R. (iv) Not injective (for example, g3(0) = g3( √ 3) = g3(− √ 3) = 0) , it is surjective, not bijective; image = R.
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