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Geometry Ch2.2 #28, 29, 30

Question from Lisa

We were working on Lesson 2-2 today and had difficulty understanding the Euler diagram in #29. As I understand it, the diagram is saying “If it is frozen carbon dioxide, then it is dry ice.” What I don’t understand is how you go from “Only if it is dry ice is it frozen carbon dioxide”to the diagram.

I’d appreciate any help you can give.

Answer from Dr Callahan

First, problem 28 has statements 1 and 2 as converse statements. We can tell that because the hypothesis and the conclusion are interchanged.

Statement 1

“If it is dry ice, then it is frozen carbon dioxide.”

is a clear “If a then b” statement with a being dry ice and b being frozen carbon dioxide.

Statement 2

has things turned around. Look at page 47 where you see “a if and only if b.” The statement in #2 reverses this. So statement 2 is the CONVERSE of statement 1.

So statement 2 says

“Only if it is dry ice is it frozen carbon dioxide.”

which is the same as

“It is frozen carbon dioxide only if it is dry ice.”

First we see that “both are true” as this is given. Now we also see they are converse statements. So now we know this is a definition.

#29 

Using the above, we can write the second statement as “It is frozen carbon dioxide only if it is dry ice.” so the diagram would have dry ice in the outside circle and frozen carbon dioxide on the inside.

#30

To combine them you get

“It is dry ice if and only if it is frozen carbon dioxide.” – A definition.