Induction

Mathematical Induction Example 7 --- Intersection of Sets

Problem: Let A₁, A₂, ..., A_n, and B₁, B₂, ..., B_n be sets. Then if A_i B_i for i = 1, 2, ..., n,   then A_i B_i .
Proof:
Basis Step: If n = 1, then A_i = A₁ and B_i = B₁ . By the hypothesis A₁ B₁ . Hence A_i B_i , for n = 1 .
Induction: Assume that A_i B_i , for an arbitrary n .
Then A_i = ( A_i ) A_n+1 by the definition of .
Since A_i B_i by the induction hypothesis, and A_n+1 B_n+1 by the hypothesis,   from property 7. of the properties of set operation ( A_i ) A_n+1 ( B_i ) B_n+1 .
Hence A_i B_i by the definition of
End of Proof