Set theory grade 9

2015.04.05

Set theory begins with a fundamental binary relation between an object $o$ and a set $A$ . If $o$ is a member (or element) of $A$ , write $o \in A$ . Since sets are objects, the membership relation can relate sets as well.

A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set $A$ are also members of set $B$ , then $A$ is a subset of $B$ , denoted $A \subseteq B$ . For example, ${1,2}$ is a subset of ${1,2,3}$ , and so is ${2}$ but ${1,4}$ is not. From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule this out, the term proper subset is defined. $A$ is called a proper subset of $B$ if and only if $A$ is a subset of $B$ , but $B$ is not a subset of $A$ . Note also that 1 and 2 and 3 are members (elements) of set ${1,2,3}$ , but are not subsets, and the subsets in turn are not as such members of the set.

Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

Union of the sets $A$ and $B$ , denoted $A \cup B$ , is the set of all objects that are a member of $A$ , or $B$ , or both. The union of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${1, 2, 3, 4}$ .
Intersection of the sets $A$ and $B$ , denoted $A \cap B$ , is the set of all objects that are members of both $A$ and $B$ . The intersection of ${1, 2, 3}$ and ${2, 3, 4}$ is the set ${2, 3}$ .
Set difference of $U$ and $A$ , denoted $U \ A$ , is the set of all members of $U$ that are not members of $A$ . The set difference ${1,2,3} \ {2,3,4}$ is ${1}$ , while, conversely, the set difference ${2,3,4} \ {1,2,3}$ is ${4}$ . When $A$ is a subset of $U$ , the set difference $U \ A$ is also called the complement of $A$ in $U$ . In this case, if the choice of $U$ is clear from the context, the notation $A c$ is sometimes used instead of $U \ A$ , particularly if $U$ is a universal set as in the study of Venn diagrams.
Symmetric difference of sets $A$ and $B$ , denoted $A △ B$ or $A ⊖ B$ , is the set of all objects that are a member of exactly one of $A$ and $B$ (elements which are in one of the sets, but not in both). For instance, for the sets ${1,2,3}$ and ${2,3,4}$ , the symmetric difference set is ${1,4}$ . It is the set difference of the union and the intersection, $(A ∪ B) \ (A ∩ B)$ or $(A \ B) ∪ (B \ A)$ .
Cartesian product of $A$ and $B$ , denoted $A \times B$ , is the set whose members are all possible ordered pairs $(a, b)$ where $a$ is a member of $A$ and $b$ is a member of $B$ . The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.

A numeral system (or system of numeration) is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.

In a positional base-b numeral system (with b a natural number greater than 1 known as the radix), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by b.

For example, in the decimal system (base 10), the numeral 4327 means (4×10³) + (3×10²) + (2×10¹) + (7×10⁰), noting that 10⁰ = 1.

Position	3	2	1	0
Weight	10³	10²	10¹	10⁰
Digit	4	3	2	7
Decimal example weight	1000	100	10	1
Decimal example digit	4	3	2	7