The And's & Or's

[lorein21]

I don't understand the rule for getting the contrapositive of condition reasoning when you have and/or.

lets say..

If I dont have B then I have H or I but not both
this means = ~B --> H or I
contrapositive = ~H + I ---> B
but theres another meaning the book suggests...

~B --> ~HI
HI --> B

this really confuses me.


also what if the not both did not exist - so it was just
if I dont have B then I have H or I.
How does it change the diagram/s?


Thank You!

-Lorein

[Eric Ockert]

Lorein

Let's start with your second question, as that is the more typical presentation you will see on the test:

"If not B, then H or I"

When you have an "or" statement, without any qualification, it implies that EITHER H or I must happen, and possibly BOTH do. To diagram this statement you would show:

~B -----> H or I

Also, the contrapositive would be:

~H + ~I ------> B

When taking the contrapositive of an "and/or" statement, flip all of your terms and negate all of your terms, as you would normally do. But you have to change any "and" to an "or, and any "or" to an "and". In this case we change the "or" from the original statement to an "and" in the second statement.


Your first example, however, eliminates the possibility of both. So if you DON'T have B, you have EITHER H or I, but NOT both. When you add the "...but not both" to the statement, that means if you don't have B, you CAN'T have BOTH H + I together. So the additional conditional statement is:

~B -----> ~ HI

and the contrapositive, "if you DO have BOTH H and I together, you must have B" :

HI -----> B

Hope that helps!

Eric