If $a = \{1,2,3\}$ and $b = \{a,b,c\},\;$ FIND $\;n(a\times b)$
Or is it impossible to multiply these sets?
What will be the answer?
$\endgroup$ 91 Answer
$\begingroup$Let's use capital letters for sets: so let $$A = \{1, 2, 3\},\;\;\text{ and} \;\; B = \{a, b, c\}$$ and $n(A) = |A| = 3,\;n(B) = |B| = 3$.
Then the Cartesian-product $\,A\times B\;$ is a set of all ordered pairs $$A \times B= \{(a_i, b_j)\mid a_i \in A, b_j \in B\}.$$
In this case, $$A \times B = \{(1, a), (1, b), (1, c), (2, a),(2, b), (2, c), (3, a), (3, b), (3, c)\}$$
In general, for two sets $P, Q$, $$\;|P\times Q| = |P| \times |Q|$$
So, if $n(A \times B) = |A\times B|,\;$ then $n(A \times B) = n(A) \times n(B) = 3 \times 3 = 9$
$\endgroup$ 10
