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Intersection of Sets


Date: 10/02/2000 at 16:09:55
From: Kristi
Subject: "And" Statements: Intersection of Sets

I do not understand intersection of sets. I need to know what it means 
and how it's used. Can you explain it in a simple, yet helpful way?


Date: 10/03/2000 at 10:29:24
From: Doctor TWE
Subject: Re: "And" Statements: Intersection of Sets

Hi Kristi- thanks for writing to Dr. Math.

Let's say we have a set of toy cars (this is our "Universal Set") and 
in our collection we have a fire truck, a red police car, a blue 
pickup truck, an ambulance, a red sports car, a white tractor-trailer 
rig and a blue station wagon.

We can make various sets out of them. Perhaps we want to organize our 
collection by color. The Red set would have: 

     R = {fire truck, police car, sports car}

The Blue set would have:

     B = {pickup truck, station wagon}

The White set would have:

     W = {ambulance, tractor-trailer rig}

We can also organize our collection into trucks and cars. The Cars set 
would have:

     C = {police car, ambulance, sports car, station wagon}

while the Trucks set would have:

     T = {fire truck, pickup truck, tractor-trailer rig}

If we wanted, we could also make a "set" out of the civil service 
vehicles. This set would have:

     S = {fire truck, police car, ambulance}

Now we have 6 different sets, all from the same collection of toys. 
Suppose our best friend, Megan, asks us, "What red cars do you have?" 
Well, our Red set (which we call R) has the fire truck, police car, 
and sports car - but they're not all cars. Our set of Cars (which we 
call C) has the police car, ambulance, sports car and station wagon 
- but they're not all red. What Megan wants to know is what vehicles 
in our collection are red AND are cars. A mathematician would call 
this the "intersection of the sets R and C." Let's see:

     R = {fire truck, police car, sports car}
     C = {police car, ambulance, sports car, station wagon}

So what vehicles are in BOTH sets?

     R^C = {police car, sports car}

I'm using the carat (^) symbol to represent intersection, or AND. Your 
book might use an upside down U or a dot instead. 

Our kid brother, Cyrus, likes vehicles with flashing lights, and he 
also likes trucks. He asks us "What trucks do you have that are civil 
service vehicles?" Well, our Truck set (which we call T) has the fire 
truck, pickup truck and tractor-trailer rig - but they don't all have 
flashing lights. Our set of civil service vehicles (which we call S) 
has the fire truck, police car and ambulance - but they're not all 
trucks. What Cyrus wants are vehicles that are civil service vehicles 
AND trucks. A mathematician would call this the "intersection of the 
sets S and T." Let's see:

     S = {fire truck, police car, ambulance}
     T = {fire truck, pickup truck, tractor-trailer rig}
So,
     S^T = {fire truck}

After we give him the fire truck to play with, Cyrus complains "I want 
a BLUE one!" So what Cyrus wants are trucks that are civil service 
vehicles that also are blue, or the "intersection of the sets S and T 
and B" in MathSpeak.

     S = {fire truck, police car, ambulance}
     T = {fire truck, pickup truck, tractor-trailer rig}
     B = {pickup truck, station wagon}

What vehicles are in ALL THREE sets? None. We call that "the empty 
set" (or the "null set"), so Cyrus is out of luck.

     S^T^B = {}

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Logic
High School Sets
Middle School Logic

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