What does the notation A × B = {(a,b) | a ∈ A and b ∈ B} mean?

[sniffer]

ehm, sorry, i am a beginner in set theory. learning on my own.
for cartesian product ordered pair, for example
$A = \{a_1, a_{2}, a_{3}\} \\<br /> B = \{b_{1}, b_{2}, b_{3}\}$

is the product $A \times B = \{a_{1}b{1}, a_{1}b{2}, a_{1}b{3}, a_{2}b{1}, \\<br /> a_{2}b{2}, a_{2}b{3}, a_{3}b{1}, a_{3}b{2}, a_{3}b{3},\}$ ??

What does $A \times B = \{(a,b)\mid a \in A and b \in B\}$ mean in detail?

thanks.

 

[sniffer]

sorry, a bit mistyped the question above.

for cartesian product ordered pair, for example
$A = \{a_1, a_2, a_3\}$ and $B = \{b_1, b_2, b_3\}$

is the product $A \times B = \{a_{1}b{1}, a_{1}b{2}, a_{1}b{3}, a_{2}b{1}, \\<br /> a_{2}b{2}, a_{2}b{3}, a_{3}b{1}, a_{3}b{2}, a_{3}b{3},\}$ ??

What does $A \times B = \{(a,b)\mid a \in A \ and \ b \in B\}$ mean in detail
in terms of individual set member for this simple example?

thanks.

 

[matt grime]

the elements in the product are the pairs (a_i,b_j) for 1<= i,j <=3.

what does a_1b_1 even mean?

the product is all odered pairs (a,b) where a is in A and b is in B. nothing more nothing less.

 

[sniffer]

$a_i$ and $b_i$ are numbers or element such as 1, 6, 8, etc.

i think i may understand your simple answer.

thanks

 

[matt grime]

the product of sets does not involve multiplying the elements; elements of sets do not necessarily even possesses a multiplicationwhat if A were the set of results of drawing a card and B were the set of results of tossing a coin? if a were the three of diamonds and b heads, then what does ab mean?

 

[HallsofIvy]

Notice the parentheses in $A \times B = \{(a,b)\mid a \in A \ and \ b \in B\}$?

What you want is
$A \times B = \{(a_{1},b{1}), (a_{1},b_{2}), (a_{1},b_3}), (a_{2},b_{1}), (a_{2},b_{2}),\\ (a_{2},b_{3}), (a_{3},b_{1}), (a_{3},b_{2}),(a_{3},b_{3}),\}$

 

[sniffer]

yup. now i understand it. thanks guys.