Operations on Sets: Explained & Examples

[bergausstein]

please help me understand what my book says:

If set A has only one element a, then $\displaystyle A\,x\,B\,=\, \{\left(a,\, b\right)\,|\,b\,\epsilon\,B\}$, then there is exactly one such element for each element from B.

can you explain what it means and give some examples. thanks! :)

 

[MarkFL]

Re: Operations on set

From Wikipedia:

In mathematics, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

For example, if the cardinality of set $A$ is one, where $A=\{a\}$ and we have a set $B$ of cardinality $n$, i.e., $B=\{b_1,b_2,b_3,\cdots,b_n\}$ then:

\(\displaystyle A\,\times\,B=\{(a,b_1),(a,b_2),(a,b_3),\cdots,(a,b_n),\}\)

 

[Deveno]

Cartesian products get their name from the prototypical example, the Cartesian plane, which is the Cartesian product of two orthogonal lines.

It's easier to see what is going on if we consider a Cartesian product of two finite sets, say:

A = a bag of red marbles,
B = a bag of green marbles.

Suppose we want "all possible pairs" of marbles, and A has 3 marbles, and B has 4 marbles. We can label these r1,r2,r3 (for the red marbles) and: g1,g2,g3,g4 (for the green marbles). Then the set of all possible pairs looks like this:

(r1,g1) (r1,g2) (r1,g3) (r1,g4)

(r2,g1) (r2,g2) (r2,g3) (r2,g4)

(r3,g1) (r3,g2) (r3,g3) (r3,g4)

Laid out like this, it's clear we have 3*4 = 12 pairs in all. And, in general:

\(\displaystyle |A \times B| = |A|\cdot|B|\)

so, if A and B are sets of 1 element each, their Cartesian product has 1*1 = 1 element (only one possible choice for the "first coordinate", and only one possible choice for the "second coordinate").