Help on the negation of statement "B union C is an independent set"

amit

New member
Joined
May 28, 2018
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Hey, was wondering if anyone here could help me with the negation of a mathematical statement, as Im not entirely sure what to do.

I wanted to negate the statement "B union C is an independent set", would the negation turn the statement into "B intersection C is a dependent set" (changing the union to intersection) or would its negation be "B union C is a dependent set" (leaving the union as is). It makes more sense to me logically that the set remains a union. But, based on how i remember doing negation (reverse everything) i should turn it into an intersection.

Thanks.
 

tkhunny

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Joined
Apr 12, 2005
Messages
10,101
Hey, was wondering if anyone here could help me with the negation of a mathematical statement, as Im not entirely sure what to do.

I wanted to negate the statement "B union C is an independent set", would the negation turn the statement into "B intersection C is a dependent set" (changing the union to intersection) or would its negation be "B union C is a dependent set" (leaving the union as is). It makes more sense to me logically that the set remains a union. But, based on how i remember doing negation (reverse everything) i should turn it into an intersection.

Thanks.
You can play with the specific language all you like. Sometimes, you will mange to make sense of it. Unfortunately, spoken language does not have the same structure as formal logical language.

In this case, the language lends itself to translation. Here is one way.

B union C is NOT an independent set

However, here my be other ways or this may be just luck. Your ONLY sure bet, EVERY TIME, is simply something like this:

IT IS NOT THE CASE THAT "B union C is an independent set".

There is logical functionality that will help dissect it after that. (DeMorgan and etc.)

Finally, you can always build a truth Table and see if what you have created is what you expected to see. Negating an existing Truth Table is very easy. Seeing if your new pieces get you to the same place can be more challenging.
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
3,698
Hey, was wondering if anyone here could help me with the negation of a mathematical statement, as Im not entirely sure what to do.

I wanted to negate the statement "B union C is an independent set", would the negation turn the statement into "B intersection C is a dependent set" (changing the union to intersection) or would its negation be "B union C is a dependent set" (leaving the union as is). It makes more sense to me logically that the set remains a union. But, based on how i remember doing negation (reverse everything) i should turn it into an intersection.

Thanks.
You are correct that IF you negate B U C (ie ~(B U C) then you get ~B Intersect ~C. But you are not negating B U C!

I agree with tkhunny that you should think: IT IS NOT THE CASE THAT B U C is an independent set which means B U C is NOT an independent set or that B U C is a dependent set.
 
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