How to list elements of the set?
I'm new to set theory and I'm having trouble figuring out this problem:
If $A= \{1,2,3\} \times \{1,2,3,4\}$, list the elements of set $B = \{(m,n)∈ A\mid m>n\}$
How would I find the elements of $B$ and would I use the Cartesian product to find them?
$\endgroup$4 Answers
$\begingroup$For each pair in B you have to take m in {1,2,3} and n in {1,2,3,4}. In this case you have to make sure that m>n, so, for example, an element of B could be (2,1), because 2 is in {1,2,3} and 1 in {1,2,3,4} and 2>1.
$\endgroup$ 0 $\begingroup$First, list the ordered pairs belonging to $A = \{1, 2, 3\} \times \{1, 2, 3, 4\}$, by making explicit this Cartesian product.
Now which one of those pairs $(m, n), \;m\in \{1, 2, 3\},\; n\in \{1, 2, 3, 4\},$ are such that $m\gt n$? The set of those pairs is $B\subseteq A$.
$\endgroup$ 1 $\begingroup$$$A= \{1,2,3\} \times \{1,2,3,4\}\\ \implies A=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4)\}$$ Since $B\subset A$, $$B=\{(2,1),(3,1),(3,2)\}$$ By the definition of $B$.
$\endgroup$ 1 $\begingroup$Make a diagram like this
4
3
2
1
*****1****2******3
You have 3 columns and 4 rows, so 3 times 4 "points" ( each point being an ordered pair) .
Create all the possible orderd pairs ( x, y) with
(1) x belonging to {1,2,3} as first element
and
(2) y belonging to {1,2,3,4} as second element.
$\endgroup$
