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Books that:

  1. has an introduction to proof, logic, and topics like sets and groups. Books

  2. can prepare you for rigorous calculus texts like Spivak and Apostol.

Question 1. I've been looking at the books by Gelfand (Algebra, Trigonometry, Functions and Graphs, The Method of Coordinates). I'm not sure if they cover all of the precalculus curriculum, though.

Question 2. What would be some good calculus books other than Spivak and Apostol?

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    $\begingroup$ I'm not sure there is such a thing as a fully rigorous pre-calculus book, since pre-calculus typically covers topics like exponentials and trigonometric functions that technically require power series, integrals, limits etc. to define properly. $\endgroup$ – Jair Taylor Jun 9 '17 at 16:14
  • $\begingroup$ @JairTaylor Are there any that emphasize problem solving, introduce proofs, and prepare you for rigorous calculus books like Spivak and Apostol? $\endgroup$ – Ansh.23 Jun 9 '17 at 16:16
  • $\begingroup$ I don't know of any good resources for this off the top of my head. You could look at this thread. $\endgroup$ – Jair Taylor Jun 9 '17 at 16:19
  • $\begingroup$ Are you inetrested in number theory or abstract algebra? $\endgroup$ – Vidyanshu Mishra Jun 9 '17 at 16:22
  • $\begingroup$ @Ansh.23, In my opinion Apostol's calculus (the two volume works) is self-contained in the sense that he never gets ahead of himself in the books. I recall that he spends some pages introducing the necessary pre-calculus stuff like summation, trigonometry, naive set theory, function theory, math induction... $\endgroup$ – Megadeth Jun 9 '17 at 16:24
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IMHO, you are not gonna find a truly amazing pre-calculus book. Just read pre-calculus topics from random sources. You can consider these books:

$1$ George F. Simmons-Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry

$2$ Michael Sullivan III -Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

For a bit higher level, consider these.

Gilbert Strang-Introduction to linear algebra.

Amann, Herbert, Escher, Joachim- Analysis $1$.

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