Abstract
In former times, mathematics was defined as the science concerned with numbers and figures. (This is reflected in the title of the classic book by Hans Rademacher and Otto Toeplitz, Von Zahlen und Figuren, literally On Numbers and Figures [6].) Nowadays, however, such a definition will not do, for modern algebra deals with abstract structures instead of numbers, and some branches of geometry study objects that barely resemble any figure in the plane or in space. Other branches of mathematics, including analysis, discrete mathematics, and probability theory, also study objects that we would not call numbers or figures. All we can say about the objects studied in mathematics is that generally, they are abstractions from the real world (but not always).
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- 1.
Jacob Bernoulli (1654–1705), Swiss mathematician.
- 2.
Fibonacci (Leonardo of Pisa) (c. 1170–1240), Italian mathematician.
- 3.
Sometimes, containment is denoted by ⊆ .
- 4.
We can write \(A_{1}\bigtriangleup A_{2}\bigtriangleup \cdots \bigtriangleup A_{n}\), since we showed in part (c) of Exercise 2.21 that different placements of parentheses define the same set (so we may simply omit them).
- 5.
That is, either prove that the identity holds for every A, B, C, or give some A, B, C for which it does not hold.
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Laczkovich, M., Sós, V.T. (2015). Basic Concepts. In: Real Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2766-1_2
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