# Thread: Need help proving!! Mathematical Induction

1. ## Need help proving!! Mathematical Induction

Hey I'm struggling in Discrete Math, could anyone help me solve this proof?

Prove that A1, A2, . . . ., An and B are sets, then (A1 ∩ A2 ∩ . . . ∩An) U B = (A1 U B) ∩ (A2 U B) ∩ . . . ∩(An U B).

This comes from the Induction and Recursion chapter of Mathematical Induction.

2. Originally Posted by Saint22
Hey I'm struggling in Discrete Math, could anyone help me solve this proof?

Prove that A1, A2, . . . ., An and B are sets, then (A1 ∩ A2 ∩ . . . ∩An) U B = (A1 U B) ∩ (A2 U B) ∩ . . . ∩(An U B).

This comes from the Induction and Recursion chapter of Mathematical Induction.
Prove it by induction for $n=2$ we need to show $(A_1\cap A_2) \cup B = (A_1 \cup B) \cap (A_2\cup B)$.
This case is proven individually.

If it is for $(A_1\cap ... \cap A_n) \cup B = (A_1 \cup B)\cap ... \cap (A_n \cup B)$.

Then if we have $(A_1\cap ... \cap A_n \cap A_{n+1})\cup B = [(A_1\cap ... \cap A_n) \cap A_{n+1}] \cup B$
And this gives, (as in the case $n=2$):
$[(A_1\cap ... \cap A_n) \cup B] \cap (A_{n+1} \cup B)$
Now apply inductive step:
$(A_1\cup B)\cap ... \cap (A_{n+1} \cup B)$
And that completes induction.