2.1.5 Guided Discovery: A dilemma in comparing quantities 1 Guided Discovery: A dilemma in comparing quantities 1. Proper subsets of a finite set a) Does one of the sets {1,2,3,...,100} and {2,4,6,...,100} contain a greater quantity of elements than the other? Solution: The first set contains exactly 100 elements, and the second set contains exactly 50 elements. Since 100 > 50, the first set contains a greater quantity of elements than the second. Notice also that both sets are finite and the second set is a proper subset of the first. What may we conclude from this? b) Would you agree with the following statement? If S is a finite set with proper subset T, then S contains a greater quantity of elements than T. Solution: This is a true statement: any finite set has a greater quantity of elements than each of its proper subsets. One can prove this using the Uniqueness Theorem on Finite Cardinalities, which states that any non-‐empty finite set has exactly k elements for a unique value of k . Conclusion: We conclude that {1,2,3,...,100} contains a greater quantity of elements than {2,4,6,...,100} because the two sets are finite and the first set properly contains the second. Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 2.1.5 Guided Discovery: A dilemma in comparing quantities 2 2. Bijections between finite sets a) Does one of the sets {1,2,3,...,100} and {2,4,6,...,200} contain a greater quantity of elements than the other? Solution: Each of these sets contains exactly 100 elements so neither contains a greater quantity of elements than the other. Notice that both sets are finite and there is also a bijection between them. What may we conclude from this? b) Would you agree with the following statement? If S is a finite set that is in bijection with a set T, then S and T contain the same quantity of elements. Solution: This is another true statement about finite sets: if there exists a bijection between two finite sets, then the sets contain the same quantity of elements. This fact also follows from the Uniqueness Theorem on Finite Cardinalities. Conclusion: We conclude that the sets {1,2,3,...,100} and {2,4,6,...,200} contain the same quantity of elements because they are both finite and there exists a bijection between them. Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 2.1.5 Guided Discovery: A dilemma in comparing quantities 3 3. We have used two facts about finite sets: Fact A: If S is a finite set with proper subset T, then S contains a greater quantity of elements than T. Fact B: If S is a finite set that is in bijection with a set T, then S and T contain the same quantity of elements. How would you answer the following? • Does one of the sets {1,2,3,...} and {2,4,6,...} contain a greater quantity of elements than the other? Solution: This question presents a dilemma: the first set properly contains the second and there is a bijection between them. Should one conclude that the first set is larger than the second, as in (1), or that the sets have the same size, as in (2)? The facts about finite sets that answered the first two questions yield contradictory results when extended to the infinite sets in (3). Without further objective reasoning, any choice of one conclusion over the other would be based on personal opinion rather than a logical argument. In Guided Discovery: More circles or more squares? you will find a valid mathematical reason for comparing sizes of infinite sets according to one view rather than the other. ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810