Cardinality issues
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues  | Cardinality of sets the cardinality of a set is its size, i.e., the number of elements in the set; we use the notation | A | to denote the cardinality of the set A |
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues  | Cardinality of sets |
| | Calculating cardinalities the "arithmetic of cardinalities" is sometimes non-intuitive, especially when infinite sets are involved
Q: What is | A È B | when A and B are disjoint (i.e., when they contain no common elements)?
Q: What is | A È B | when A and B are not disjoint? |
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues | | Cardinality of sets |
| | Calculating cardinalities |
| | Infinite cardinalities the "arithmetic of cardinalities" is especially non-intuitive when infinite sets are involved; in general, we may recognize several "orders" of infinity |
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues | | Cardinality of sets |
| | Calculating cardinalities |
| | Infinite cardinalities |
| | Countable vs. uncountable we say that a set is countably infinite if its members can be put in one-to-one correspondence with the natural numbers
we say that a set is uncountable if its members can not be put in one-to-one correspondence with the natural numbers
Q: what is the cardinality of A È B, where A and B are disjoint, countably infinite sets? |
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues | | Cardinality of sets |
| | Calculating cardinalities |
| | Infinite cardinalities |
| | Countable vs. uncountable |
| | Cardinality of products for finite sets, the cardinality of a product is the (arithmetic) product of the two cardinalities
Q: what is the cardinality of the product of two countably infinite sets? |
| Lecture #2: Review of Relevant Mathematics |
Cardinality issues | | Cardinality of sets |
| | Calculating cardinalities |
| | Infinite cardinalities |
| | Countable vs. uncountable |
| | Cardinality of products |
| | Cardinality and powersets for finite sets, the cardinality of the powerset is 2 to the power of the cardinality of the set (why?)
for infinite sets, the cardinality of the powerset is strictly larger than that of the base set |