I am a newbie to Stack-Exchange and if there is any problem in my question -- I apologize beforehand .

I was working my way through some Propositional Logic Questions in Discrete Maths by Rosen , when I came across the following question :

Let p, q, and r be three propositions

p : Grizzly bears have been seen in the area

q : Hiking is safe on the trail.

r : Berries are ripe along the trail.

Write the propositions using p, q, and r and logical connectives (including negations):

For hiking on the trail to be safe, it is necessary but not sufficient that berries not >be ripe along the trail and for grizzly bears not to have been seen in the area

My Solution :

q => (~r AND ~p) -- because (~r AND ~p) is the necessary condition

Book's Solution :

(q => (~r AND ~p) ) AND ~((~r AND ~p) => q)

Doubt :

I am puzzled why the book's solution is as it is given .

Can someone help me out ? Would be grateful .

  • 2
    Are you sure that that is the book's answer and not "(q => (~r & ~p)) & ~((~r & ~p) => q)"?
    – Geoffrey
    Dec 21, 2014 at 15:57
  • 1
    The question made my day - I had a really good laugh :)
    – user132181
    Dec 21, 2014 at 19:10

2 Answers 2


The left-hand side that you translate captures that it is necessary. The right-hand side the book also provides is that it is not sufficient. The "English" states both.

  • Hi , virmaior -- but doesn't the fact that the condition is not sufficient is captured in the fact that (~r AND ~p) comes on the right hand side of the implication ?
    – pranav
    Dec 21, 2014 at 6:38
  • 1
    No. That leaves it ambiguous whether it is or is not sufficient. Because it is equally compatible with the book's answer and a biconditional rather than conditional (i.e. it is compatible with it being both necessary and sufficient).
    – virmaior
    Dec 21, 2014 at 6:41

I think the problem is poorly formulated (not your fault of course). The author puted the cumbrous ideas of "necessity" and "sufficience", which puts us into epistemology, when a simple class-calculus would do the trick, being at the same of much more use for the working mathematician. I.e, we take

enter image description here enter image description here enter image description here

The proposition you wnat is:

enter image description here

I.e, those who hike on the trail T, in the area A, are safe, when the class of all berries in this trail is not empty and when the class of all bears saw in the area A is empty; or, when the class of all berries in the trail T is not empty, and when the class of all bears saw in the area A is empty, then everyone who walks on T is safe.

For the Russell-Peano notation and the basics of class-calculus, see https://archive.org/stream/PrincipiaMathematicaVolumeI/WhiteheadRussell-PrincipiaMathematicaVolumeI#page/n49/mode/2up

Have a nice day.

  • 1
    @pranav You're welcome buddy. I notice you're interested in discrete mathematics, a very nice subject. But unfortunately you're reading from the worst source you can find..Rosen's book doesn't make the distinction, for example, between 1 and +1, although this difference is fundamental for the theory of progressions, and a whole bunch of other mistakes that popular math books often do (regarding fractions, relations, etc.). If you're interested send me an email (grupodemonodromia@gmail.com) and I'll pass you the best authors on the basics (relations, integers, etc.), all available online.
    – Ricardo
    Jan 2, 2015 at 13:30
  • Hi , @Ricardo - thanks for the offer :) . Maybe you can list out the books in the comments here or better - make a new question / mega list on Math.SE of the books so that others too can benefit from your knowledge :)
    – pranav
    Jan 3, 2015 at 4:52
  • 1
    @pranav I can post as a comment if you want, but I won't do it on math SE. I had some experience with it, discussing philosophy of mathematics, and I can tell you they all are followers of Zermelo and Gödel, mediocre, who thinks a fraction is the same as a division, who makes no distinction between a and +a, although even the most elementary algebra book does it. They deny Russell's approach on the logic of relations, although their definition on integer and fraction is obviously based on relations. Anyway, they'd probably close the quest. I.e, I made some enemies there.Still prefer the email!
    – Ricardo
    Jan 3, 2015 at 13:07

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