How Do You Prove Subset Relationships Within Intersecting Indexed Sets?

[gbean]

Homework Statement

Show that the intersection of Ai (for all i in I = {1, 2, 3, ... n } = A1. Ai is a subset of Aj whenever i <= j.

Homework Equations

The Attempt at a Solution

Show:
***I'm having trouble showing part 1***1. that the intersection of Ai is a subset of A1, and
2. A1 is a subset of the intersection of Ai.

This is my attempt: 1. Let x be an element of the intersection of Ai. Then x is in Ai for all i in I. Since A1 is contained in all Ai, then x is contained in A1.

2. Let x be an element of A1, then as A1 is a subset of Aj, for all j >= 1, x is an element of Aj. Thus, x is an element of the intersection of Ai.

 

[Dick]

I think that's completely correct. Except maybe that you don't need A1 is a subset of Ai for the first part. If x is in the intersection of the Ai, it's certainly in A1.

 

[SammyS]

"This is my attempt: 1. Let x be an element of the intersection of Ai. Then x is in Ai for all i in I. [STRIKE]Since A1 is contained in all Ai, then x is contained in A1.[/STRIKE]"

Since x ∈ A_i for all i ∈ I, then clearly, x ∈ A₁, because 1 ∈ I .

(Not that what you had was incorrect, but I think this is more direct.)

You could do (2.) by induction.