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Numbers, trigonometric functions and coordinate geometry
Published in Alan Jeffrey, Mathematics, 2004
If a set only contains a small number of elements it is often simplest to define it explicitly by enumerating them. Hence, for a set S comprising the four integer elements 3, 4, 5 and 6 we would write S = {3, 4, 5, 6}. Whereas this set is a finite set, in the sense that it only contains a finite number of elements, the set N is seen to be an infinite set.
Exploring main obstacles of inflexibility in mathematics teachers’ behaviour in accepting new ideas: the case of equivalence between infinite sets
Published in International Journal of Mathematical Education in Science and Technology, 2019
A set X is infinite if there exists one-to-one correspondence (a bijection map) between X and some proper subset of X (Dedekind infinite). In equivalence between two sets, finite sets with the same size are equivalent by a one-to-one correspondence between all their elements. There are some complexities to generalize this pattern to infinite sets; because, based on Dedekind definition, an infinite set can be equivalent with some of its proper subsets while it is impossible in finite sets. According to Barahmand [14], overgeneralization of the known finite results to infinity causes problems for many students. The property of infinite sets that we can add or subtract a finite number of elements and their sizes remaining the same is one of the overgeneralizations. Because, this is not the case for finite sets and as Tirosh [15] quoted from students’ viewpoint, a set and its proper subset cannot be equivalent. This criterion which, a proper subset of a set has fewer elements of the set itself, is one of the current criteria used by individuals regarding the comparison of infinite sets called part-whole.
Likert-scale questionnaires as an educational tool in teaching discrete mathematics
Published in International Journal of Mathematical Education in Science and Technology, 2018
O. A. Ivanov, V. V. Ivanova, A. A. Saltan
Statements of questionnaire 1 (Here statements 2, 3, 6, 8, and 9 are the true ones.) The magnitude of a Fibonacci number is a quadratic function of its index.The set of all arithmetic progressions is the same as the set of all sequences satisfying a second-order linear recurrence relation.The number of combinations is a special case of the number of permutations with repetition allowed.The sum of the squares of the first n natural numbers is a quadratic polynomial in n.If the height of a tower of Hanoi is doubled, then the number of moves required to transfer the entire stack to another peg is approximately four times greater.If finite sets A and B have the same number of elements, then the number of injections A → B coincides with the number of surjections A → B.In every set, the number of all 4-element subsets is greater then the number of all 3-element subsets.If a set has more than 3 elements, then the number of permutations on its elements is greater than the number of its subsets.The number of subsets of a finite set A coincides with the number of all maps from A to a 2-element set.If 13 out of 25 schoolmates like to read books and 17 adore computer games, then exactly 5 of them like both to read books and play on the computer.