teaching:topics:number:axioms-formal
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teaching:topics:number:axioms-formal [2024/05/02 09:40] simon |
teaching:topics:number:axioms-formal [2025/02/24 06:54] (current) simon |
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| We can make interesting and useful sets of things where only some of these properties hold or we can | We can make interesting and useful sets of things where only some of these properties hold or we can | ||
| make finite sets of things that otherwise have all the properties of, say, the integers or rationals. | make finite sets of things that otherwise have all the properties of, say, the integers or rationals. | ||
| - | At school we are mostly looking at either whole numbers (dealing with **discrete** things --- that we can count) or fractions and real numbers (dealing with **continuous** things --- quantities we can measure like length or weight). The difference | + | At school we are mostly looking at either whole numbers (dealing with **discrete** things --- that we can count) or fractions and real numbers (dealing with **continuous** things --- quantities we can measure like length, weight and angles or ratios, scales and proportions). The difference |
| between these two kinds of number is profound, and starting to understand that difference is a crucial step | between these two kinds of number is profound, and starting to understand that difference is a crucial step | ||
| in school maths. | in school maths. | ||
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| ===some set notation, as used here ...=== | ===some set notation, as used here ...=== | ||
| * a set is a collection of distinct elements, `a=b` means they both represent the same element | * a set is a collection of distinct elements, `a=b` means they both represent the same element | ||
| - | * `\ b in ccN quad` means: we are giving the name `b` to "an element of" (that is something "in") the set which we are calling `ccN` | + | * `\ b in ccN quad` means:`quad`we are giving the name `b` to "an element of" (that is something "in") the set which we are calling `ccN` |
| * this is exactly what we do when we use a pronumeral in algebra | * this is exactly what we do when we use a pronumeral in algebra | ||
| - | * `forall b in ccN quad` then means: for all `b` in `ccN` | + | * `forall b in ccN quad` then means:`quad`for all `b` in `ccN` |
| * in other words:`quad`"for every possible `b` that we could choose from `ccN` ..." | * in other words:`quad`"for every possible `b` that we could choose from `ccN` ..." | ||
| - | * `exists c in ccN quad` means: there exists something we will call `c` in `ccN` | + | * `exists c in ccN quad` means:`quad`there exists something we will call `c` in `ccN` |
| * in other words:`quad`"we can always find some suitable element `c` in this particular collection called `ccN` ..." (usually with some property that will be given given next). | * in other words:`quad`"we can always find some suitable element `c` in this particular collection called `ccN` ..." (usually with some property that will be given given next). | ||
| - | * `quad : quad` means "such that" | + | * `quad : quad` means`quad`"such that" |
| * it can often be read as "given" or "where" | * it can often be read as "given" or "where" | ||
| * in other words:`quad`'... with the following conditions ...' | * in other words:`quad`'... with the following conditions ...' | ||
| - | * so rule \(\eqref{multiply closed}\) ... `quad forall a, b in ccN qquad exists c in ccN : quad c = a × b`\\ reads as:`quad`for every `a` and `b` in `ccN` there is some `c`, also in `ccN`, where `\ c = a × b`. | + | * so rule \(\eqref{multiply closed}\) ...`quad forall a, b in ccN qquad exists c in ccN : quad c = a × b`\\ reads as:`quad`for every `a` and `b` in `ccN` there is some `c`, also in `ccN`, where `\ c = a × b`. |
| * "whenever we multiply two of these numbers the result is also one of these numbers" | * "whenever we multiply two of these numbers the result is also one of these numbers" | ||
| - | * this is what we mean when we say: "`ccN` is //closed// under multiplication" | + | * this is what we mean when we say:`quad`"`ccN` is //closed// under multiplication" |
| * some logic operations: \(\ \therefore\,\implies \land\ \lor\ \neg \quad\)mean: therefore, implies, and, or, not. | * some logic operations: \(\ \therefore\,\implies \land\ \lor\ \neg \quad\)mean: therefore, implies, and, or, not. | ||
| * some more set notation: \(\ \cap\ \cup \subset\ \emptyset\ \notin \quad\)for: intersection, union, subset, the empty set, not in | * some more set notation: \(\ \cap\ \cup \subset\ \emptyset\ \notin \quad\)for: intersection, union, subset, the empty set, not in | ||
teaching/topics/number/axioms-formal.1714606816.txt.gz · Last modified: 2024/05/02 09:40 by simon
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