Demorgan Law: Help

mcmlaw36

Just some Confusion on statement with both "and" and "or" in the stimulus. For example how would I draw a conditional statement with a stimulus like "If A or B than C and D" and ho would you negate this statement as well. may be a dumb question but having trouble drawing it out, especially while splitting the 'or" in the Sufficient and the "and" in the necessary

Accounts Playable

It translates like this: "(A or ---->(C and D)" One of A or B happening is sufficient to trigger both C and D.

Negating the a conditional statement means that you can have the sufficient condition trigger, but the necessary condition not trigger.

So, (A or can happen while (C and D) doesn't happen, or translated into lawgic:
(A or SOME Not (C and D).

Think of an intuitive example: At the store, I can buy apples or broccoli while also not buying both carrots and detergent.

runiggyrun

The rule of thumb is that you can split "OR" in the sufficient and "AND" in the necessary.
For "if A or B than C and D" you have two steps
Step one - split the sufficient: either A or B happening triggers C and D.
So:
A-->C+D
B-->C+D

Step two - split the necessary: A triggering C and D can further be split into
A-->C
A-->D
Same for B.

The fully split statements are
A-->C
A-->D
B-->C
B-->D

The contrapositive for the whole statement would be
If not C OR not D then not A AND not B (negate and reverse, swap AND and OR)
Again, you can split the OR in the sufficient and the AND in the necessary
Split the sufficient:
/C-->/A+/B
/D-->/A+/B
Split the necessary:
/C-->/A
/C-->/B
/D-->/A
/D-->/B

Notice that you end up with the same 4 contrapositives whether you negate the original statement and then split, or you split the original statement and then negate.

mcmlaw36

Is it valid to say. (not) C or D ----> (not) A and B ?

Accounts Playable

@mcmlaw36

I think you're confusing the negation and contrapositive.

The contrapositive (flip and negate) of (A or --->(C and D) is Not (C and D)---> Not (A or . This simplifies down to:
(Not C or Not D)--->(Not A and Not . The original statement and the contrapositive have the exact same meaning; both are restatements of each other using different words.

The negation of a conditional statement denies the truth of the conditional relationship. So, if you deny the conditional statement, you are saying "It isn't true that (A or --->(C and D)," which means (A or is not sufficient for (C and D). It can also mean that (C and D) is not necessary for (A or .

Q.E.D

Seems like folks have trouble when they think about logic in terms of rules instead of meaning.

Yes, there's a rule saying
/(P or Q) = (/P and /Q).

But it's no use trying to remember all the rules. If you think of the logical meaning of a statement, you don't need to remember the rules bc they'll be obvious.

Knowing the meaning of a statement is knowing how it can be true or false.

How can it be true that P or Q? Well, either P or Q (or both) being true would make it true that 'P or Q'. If that's so, then how could it be false? If either one is true, then the whole statement is true, so both P and Q must be false to render 'P or Q' false.

Ergo,
/(P or Q) = (/P and /Q)

The same thought process, simple as it sounds, is the right approach to logical reasoning. You can't be good or fast at logic until you switch from memorizing axioms (rules) to interpreting meaning.

I suggest you look up DeMorgan's laws and, for each one, cover one part and try to guess the other by thinking about the meaning of the side you can see. Do it as a warm-up every day. Make sure you understand the material conditional (when it's true and when it's false). When you're comfortable with both, you'll have little difficulty combining them in more complex sentences like the one you're asking about.

In response to your question:

Contrapositive:
(/C or /D) implies (/A and /B)

Negation:
(A or and (/C or /D)

Note: The "negation" of the conditional is synonymous with the "contrapositive" in the popular vocabulary, but these are different things. The negation of a conditional is the condition that makes the conditional relationship false. The contrapositive represents the modus tollens inference when the conditional is true but the consequent is false.

Good luck!

runiggyrun

@mcmlaw36 no, you can't say "(not) C or D ----> (not) A and B".
I skipped this in my original explanation, but the very first step of the contrapositive for a compound conditional is the same as for any conditional : simply negate and reverse.
For A or B--> C and D, negate and reverse gives you
Not (C and D) --> Not (A or .
The next step is applying the "not" to what's in the parantheses, and that's where DeMorgan law comes in
Not (C and D) becomes Not C OR Not D (AND in parantheses turns to OR when split)
Not (A or becomes Not A and Not B. (OR in parantheses turns to AND when split)

MrSamIam

Rule of thumb: With and or, or in a conditional statement, you contrapose by flipping, negating, and switching the "and" to an "or," or the "or" to an "and."

So, for your statement:
A or B --> C and D Note: The A and B can split since both are independently sufficient to lead to C and D (in other words, as soon as I see EITHER A or B, I know that BOTH C and D will occur).
Technically the C and D can also split, but in this case I would not split them. You've already split the "A or B" statement and splitting both sides of the conditional chain will cause confusion.

Contrapositive:
(not) C OR (not) D ---> (not) A AND (not) B
The same rule applies for splitting. C or D are both INDEPENDENTLY sufficient for cause both A and B. Again, you would just split the arrows coming off of A/B and leading to "C and D"

shizuokatwin379

I just use OLAR , or left, and right. Makes it super easy for me at least to remember which splits. I mean it really does make common sense if you think about.it, but the little anagram really helps for quick changes in LGs.

To do demorgans I just reverse the sufficient and necessary (including and/or) then apply the rule.