The Math Profs
Hello \( 9 + 9 \) Hello
\[ x ^ 2 \]
\( \bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{a_i\}\times\{b_j\}\right]= \bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{(a_i,b_j)\}\right] \)
Let \(A=\{a_1,a_2,\ldots\} \) and \(B=\{b_1,b_2,\ldots\}\)
\[ A \times B
=\left(\bigcup_{j=1}^\infty A\times\{b_j\}\right )
=\bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{a_i\}\times\{b_j\}\right]
=\bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{(a_i,b_j)\}\right] \]
\[ A \times B
=\left(\bigcup_{j=1}^\infty A\times\{b_j\}\right )
=\bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{a_i\}\times\{b_j\}\right]
=\bigcup_{j=1}^\infty\left[\bigcup_{i=1}^\infty\{(a_i,b_j)\}\right] \]
