1. Show that the size of the set of all positive integer multiples of 7 has cardinality No. 2. Show that the size of the set of all positive integer multiples of k has cardinality No for any k € Z+. 3. Show that the size of the even integers is the same as the size of the set of all positive integer multiples of 5. 4. Let A = {1,2}. Show that |A| ‡ |A × A|.
Background info:
Here, this covers the concept of different sizes of infinity. Afterall, when sets are finite, the answer is easy since the size of a finite set is a number. The order of a set A, |A|, is the number of elements it contains. Let A = {1, 2, 3}, B = {q, f, z} and C = {♠, ◇} Note that |A| = 3 = |B| so A and B have the same size. Since |C| = 2, A and C have different sizes. When sets are infinite, things get trickier since ထ is a concept rather than a number. So, We need a different approach that is still consistent with the concept of size of finite sets. A sturdier definition that works with both finite and infinite sets is to say that two sets have the same size if there exists a one-to-one and onto function between the sets. Note that one-to-one and onto functions are invertible. Hence order is a symmetric relation. As such, A and C have different sizes since we cannot map all three elements of A to C with a one-to-one function. Conversely, if we attempt to map C to A, no onto function exists. This approach works with sets of infinite size. We define the cardinality of the positive integers as countably infinite. Symbolically, |Z+| = ℵ0 (aleph null)
What I am not understanding is what makes a function one-to-one and/or onto? Also how would we apply the given background information to prove the following problems? (attached as images)
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