Question

please help me with the following question thank you

1. Suppose that A is a set and {B_i | i ∈ I} is an indexed families of sets. Prove that
   A × (U_i∈I B_i) = U_i∈I (A × B_i).

This problem is similar to Theorem 4.1.3 and to Exercise 11 in Section 4.1 of your SNHU MAT299 textbook.

Answer 1

Let

(x, y)E A x B

(a A) A (y EB

(x E A) A (y E B1 Vy E Bi Vy E Big V 21, 22, 13... E I 2

Η ιαΕ ΑAYE BVα E ΑΛyE BVαΕ ΑA Β)V , 2, 3 . ΕΠ

r, y) E A x Bi V (, y) E A x Bi V (, y) E Ax Big V. 1, i2, i3 E I

$\small \Leftrightarrow (x,y) \in \bigcup _{i \in I}(A \times B_i)$

Hence  

U(A x B) LA x Bi iEI