What Happens When You Intersect Before or After a Cartesian Product? | Sets | Discrete Math | Dgmthc

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What happens when you take the intersection of two sets and then form a Cartesian product with a third set? Does it matter whether you intersect first and then multiply, or multiply first and then intersect? In this video we prove that it doesn't matter at all. The result is exactly the same either way.
Formally, we prove that for any sets A, B, and C:
(A ∩ B) × C = (A × C) ∩ (B × C)
This is one of those results in set theory that looks a little intimidating at first glance, but once you understand what each side is actually saying, the proof becomes surprisingly natural. We walk through it multiple ways so you can build real intuition, not just symbol-pushing.
First, we lay out the proof strategy. To prove two sets are equal in mathematics, the standard approach is to show that each set is a subset of the other. This is called the double containment method, and it's one of the most important proof techniques in all of discrete mathematics. We need to show:

(A ∩ B) × C ⊆ (A × C) ∩ (B × C) ... and ...
(A ∩ B) × C ⊇ (A × C) ∩ (B × C)

Before diving into the algebra, we build visual intuition using Cartesian coordinate grids. We plot the sets on axes and show what each side of the equation literally looks like as a region in the plane. Spoiler: both regions are identical rectangles. The picture makes the proof feel obvious, which is exactly the point.
Then we write the formal proof. The technique we use is called element chasing, we assume an arbitrary ordered pair (x, y) belongs to one side and carefully trace which sets x and y must belong to, using only the definitions of Cartesian product and intersection. We do this in both directions, proving each containment rigorously.
Finally, we show a cleaner version of the proof using a biconditional chain, a sequence of if-and-only-if steps that handles both directions simultaneously. Read it top to bottom for the forward direction. Read it bottom to top for the reverse. One chain of double arrows, proof complete.


Properties and Concepts Used:
Cartesian Product — Given sets X and Y, the Cartesian product X × Y is the set of all ordered pairs (a, b) where a ∈ X and b ∈ Y. For example, if X = {1, 2} and Y = {3, 4}, then X × Y = {(1,3), (1,4), (2,3), (2,4)}.

Set Intersection — A ∩ B is the set of all elements that belong to both A and B at the same time. If x ∈ A ∩ B, then x ∈ A and x ∈ B.

Ordered Pair — A pair (x, y) where the order matters. (x, y) is not the same as (y, x) unless x = y.

Double Containment Proof — The standard method for proving two sets are equal: show A ⊆ B and B ⊆ A, which together imply A = B.

Element Chasing — A proof technique where you assume a generic element belongs to one set and derive, step by step using definitions, that it must also belong to another set.

Biconditional (⟺) — A logical statement that holds in both directions. "P ⟺ Q" means P implies Q and Q implies P. A chain of biconditionals is a powerful way to write a two-directional proof in one pass.

Subset (⊆) — We say A ⊆ B if every element of A is also an element of B.

Chapters:
0:00 Introduction
0:45 Visual Proof: Cartesian Grids
1:26 Formal Proof: Forward Direction
2:55 Formal Proof: Reverse Direction
4:15 Biconditional Chain: A Simpler Proof
5:12 Other Cool Stuff and Thanks for Watching


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