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Proofs and Concepts: The Fundamentals of Abstract Mathematics

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Figure 6A. Arrow diagrams of three functions f, g, and h.
Figure 6B.f is onto, but not one-to-one.
Figure 6C. Arrows for the composition g ◦ f are dotted.
Figure 7A. A partition of A into subsets A1, . . . , A5. (Each element of A is in oneand only one of the subsets.)
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5.3 Give a two-column proof of the deduction 1.2 Suppose A and B are sets. 2.12 Suppose that p is a constant, but all other lower-case letters represent variables For each of the following, (a) does it have a free variable? (b) is it an assertion? 3.8 Which of the following English assertions are vacuously true (in the real world)? 1.27 Let n ∈ Z

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