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text: '[summary: The product of two [3jz sets] $A$ and $B$ is just the collection of ordered pairs $(a,b)$ where $a$ is in $A$ and $b$ is in $B$. The reason we call it the "product" can be seen if you consider the set-product of $\\{1,2,\\dots,n \\}$ and $\\{1,2,\\dots, m \\}$: it consists of ordered pairs $(a, b)$ where $1 \\leq a \\leq n$ and $1 \\leq b \\leq m$, but if we interpret these as integer coordinates in the plane, we obtain just an $n \\times m$ rectangle.]\n\n[summary(Technical): The product of [3jz sets] $Y_x$ indexed by the set $X$ is denoted $\\prod_{x \\in X} Y_x$, and it consists of all $X$-length ordered tuples of elements. For example, if $X = \\{1,2\\}$, and $Y_1 = \\{a,b\\}, Y_2 = \\{b,c\\}$, then $$\\prod_{x \\in X} Y_x = Y_1 \\times Y_2 = \\{(a,b), (a,c), (b,b), (b,c)\\}$$\nIf $X = \\mathbb{Z}$ and $Y_n = \\{ n \\}$, then $$\\prod_{x \\in X} Y_x = \\{(\\dots, -2, -1, 1, 0, 1, 2, \\dots)\\}]\n\n[todo: define the product as tuples]\n\n[todo: several examples, including R^n being the product over $\\{1,2, \\dots, n\\}$; this introduces associativity of the product which is covered later]\n\n[todo: product is associative up to isomorphism, though not literally]\n\n[todo: cardinality of the product, noting that in the finite case it collapses to just the usual definition of the product of natural numbers]\n\n[todo: as an aside, define the product formally in ZF]\n\n[todo: link to universal property, mentioning it is a product in the category of sets]',
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