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4.0 out of 5 stars Clear, "logical" introduction, 29 July 2013
C. R. Long "clive_long" (London) - See all my reviews
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This review is from: Elements of Logic via Numbers and Sets (Springer Undergraduate Mathematics Series) (Paperback)
This book clearly lays out the language and techniques of logic applied to material familiar to a UK A-Level or First-year undergraduate student; numbers and series. It then introduces potentially unfamiliar key ideas such as well-ordering, partitioning of sets and equivalence relations. The book ties together logic and sets so the student can see why the algebra of sets is valid rather than some plausible results based on Venn diagrams. I especially liked how the key techniques for proof : modus ponens, modus tollens, proof by contradiction and proof by contraposition were expressed using logic operators then shown to yield tautologies by truth tables. Good mathematical style is introduced by worked examples that the student can follow and try to emulate.

The exercises are of a suitable standard and presented regularly throughout the text. I really felt it was important to tackle the exercises as they arose to check whether I had understood (or not) the preceding material. There are solutions to most exercise questions. The book , like other Springer-Verlag texts, is reasonably priced for most students and clearly organised and printed.
For me the material on syllogisms was less successful and less easy to follow, but maybe that is me.

I still think the gold-standard for an accessible yet exhaustive treatment of logic applied to proofs is “How to Prove it” by Velleman. That book does not yield its rewards instantly but after working through the exercises and really coming to grips with the material you really feel you understand (or begin to understand) what a valid argument actually is.
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