# 34 is a rational number 2 is not a rational number

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Unformatted text preview: set of positive even integers E is a countable set. Solution: let f (x) = 2x. Z+: 1 2 3 4 5 6 … 2 4 6 8 10 12 … Then f is a bijection from Z+ to E since f is both one- to- one and onto. Posi\$ve Ra\$onal Numbers are Countable ༉  Reminder: a rational number can be expressed as the ratio of two integers p and q such that q ≠ 0. ༉  3/4 is a rational number. ༉  √2 is not a rational number. Example 2: show that the positive rational numbers are countable. Solution: the positive rational numbers are countable since they can be arranged in a sequence: r1 , r2 , r3 , … The next slide shows how this is done. → Posi\$ve Ra\$onal Numbers are Countable First row q = 1. Second row q = 2. etc. Constructing the List First list p/q with p + q = 2. Second list p/q with p + q = 3 Third list p/q with p + q = 4, and so on. Matrices ༉  Matrices are useful discrete structures that can be used in many ways. For example, they are used to: ༉  Describe certain types of functions known as linear transformations , e.g., they are used in computer graphics. ༉  Express which vertices of a graph are connected by edges. ༉  Later, we will see...
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