Saturday, August 4, 2018

Complement (set theory)


FTR  Concept of a 'complement':

If A is the area colored red in this image... ... then the complement of A is everything else.
 
 In set theory, the complement of a set A refers to elements not in A. 
When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. 
The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A.

If A is the area colored red in this image...

... then the complement of A is everything else.



Now via an Art methodology:





21 comments:

Bob said...

Negation, negation, negation.

Bob said...

Did you know that the complement of Democrats and Republicans is all that is decent?

Tom Hickey said...

Did you know that the complement of Democrats and Republicans is all that is decent?

Not to argue with this, but actually, the political complement of opposing positions is compromise.

The comprise may not be "decent" however. It may just be compromise between factions of an oligarchy.

But the idea is that in a genuine democracy as government of, by and for the people, compromise will lead to social optimizing dynamically that is, over time with conditions changing.

But today in the US there is no genuine democracy, just the jockeying of elites with the different parties representing different elite factions.

Bob said...

Google the "Republican Democrat Venn Diagram" and you'll see what many people have in mind. By sheer coincidence, the overlap between the two includes the important stuff.

Tom Hickey said...

The definition of set and complement above in the post is instructive.

This is the idea of a thesis excluding the anti-thesis. Thus they are mutually exlcusive,

The fact that there may be a synthesis is therefore ruled out.

But if the two sets are represented in a Venn diagram it is possible that they overlap and to that degree are complementary.

This is generally the case with compromise and in the historical dialectic, one moment being followed by another moment that incorporates a portion of the path, since history is path-dependent.

For example, what we are now witnessing in the case of China is a complement (compromise) growing out of the clash between capitalism and communism, liberalism and traditionalism.

Thus, China is now on the cutting edge of the historical dialectic. Russia, also, but to a different degree, since conditions are different there.

This was presaged in the US in the New Deal and in Europe with the rise of social democracy.

Bob said...

Isn't the above diagram a Venn diagram?
With one set + U, the absolute complement and the relative complement are the same.
All other sets other than U are subsets of U.

Dems and Repubs have too much in common to be considered opposing political forces. If a statistical weight were given to each issue and converted into area, the intersect might be a near eclipse.

Tom Hickey said...

You are claiming that GOP and Dems are the same set and the compliment is empty?

I would say two set with major differences and some overlap regarding elite priorities like military spending.

Bob said...

Take this one as an example:
https://www.pinterest.ca/pin/178244097731404311/

Bob said...

Or this:
https://chaos.social/@pinkprius/99140106198559261

Matt Franko said...

Bob the Venn is presented as an intersection of 2 sets ... complement is a division of a whole into sub sets...

Iow the Venn does not recognize the whole set...

Like “supply and demand!” doesn’t recognize the basic underlying surplus condition..

Iow you have mankind the whole divided into complementary sets of political parties that make up the whole..

This opposed to how it’s done dialectically where there are just two adversarial sets and the whole is gone from the scene...

Matt Franko said...

In the diagram the whole is represented by the square outer boundary...

Tom Hickey said...

Dialectic assumes many sets with some overlapping. The empty space in in the square is the uncertain future where new sets will appear dynamically, either through discovery or innovation in the field of possibility (knowledge) of or through interaction on the field of actuality (action, history).

This is explained at the outset of Hegel's Logic. Positing being calls for nothing as the compliment of being and the opposition of being and nothing results in becoming. The categories develop dialectically out of the dialectic of thought.

This approach to logic is applied to history in the rest of Hegel's work on philosophy, and magisterially in the Phenomenology.

It's not possible to understand dialectic without grasping Hegel's Logic and its concretization in the Phenomenology.

Without understanding this it is not possible to know what Marx was doing.

And it is not possible to understand what Hegel was doing without understanding Kan't three critiques, but the Critique of Pure Reason in particular.

Plus, there was a transition from Kant to Hegel through the German idealists that followed Kant, Fichte and Schelling in particular.

And scholars argue over the particulars of this.

So thinking that one can just read some blurb on dialectic and understand it in term of thesis > anti-thesis > synthesis is simplistic.

Standing in contrast to Hegel's logic is the contemporary formal logic aka mathematical logic and symbolic logic. Formal logic is a developed and higher math and just as complicated and challenging.

Economics and social science are attempts to apply formal logic and math to real-world problems.

Thus the dialectical approach and the formal approach are contrasting and also complimentary.

Bob said...

The box represents the universal set, the set of all sets, including the empty or null set. Is that not 'the whole'?
You can ignore the whole by not drawing the box. Or you label the box with "US Political Parties". Or you assume that there aren't an infinite number of sets.

In the real world, these situations are known as dichotomies. Some of them turn out to be false upon examination. Democrats versus Republicans is a false dichotomy.

Tom Hickey said...

Democrats versus Republicans is a false dichotomy.

One would not conclude that after looking at the polling.

There are extreme differences involving an overwhelming number of the populations of each set, and also some overlap, generally accountable based on agreement among elite factions of the parties.

Tom Hickey said...

Venn diagrams are drawn in possibility space, which contains all possible sets.

Sets are defined by characteristics. Venn diagrams represent the sets under consideration. The complement is all the sets in possibility space that have not been selected for comparison.

Wrt to the sets selected for comparison, possibility space is stipules to be empty of members that don't fit into those sets based on the defintions. This is all based on stipulations. The stipulations may or may not reflect distributions in actual space. That is a question that requires empirical investigation.

Sets can be viewed as concepts that abstract from non-essentials, which the concept excludes or "brackets." In other words, set member may have other characteristics but they are abstracted from for simplification.

Sets may be either either sharp or fuzzy. Classical logic/set theory generally assumes sharp sets. Fuzzy logic deal with fuzzy sets.

Matt Franko said...

“understand it in term of thesis > anti-thesis > synthesis is simplistic.”

Yes but nevertheless that seems to be what is being done...

Same as we know we’re not “out of money!”’yet everybody else is going all around saying “we’re out of money!”...

Just because WE know something doesn’t mean everybody else does...

Bob said...

The Venn diagrams I linked to list policies. This is consistent with a study looking at the level of influence that the public has on policymaking.

Tom Hickey said...

The Venn diagrams I linked to list policies. This is consistent with a study looking at the level of influence that the public has on policymaking.

Agree, but I don't see how one can extend this to parties as a whole without first qualifying it. Maybe you did and I missed it?

Bob said...

I don't know what you mean. Which policies do Democrats and Republicans agree on? Also expressed as bipartisan agreement. Military spending would be an example.

The party platforms would have a greater number of policies, which in practice are not voted on, or are committee'd into limbo.

The diagrams are looking at those policies that get voted on and are passed. There's also an inference that what doesn't get passed is due to mutual consent. If those items were listed, it would be made more explicit.

Tom Hickey said...

I don't know what you mean. Which policies do Democrats and Republicans agree on? Also expressed as bipartisan agreement. Military spending would be an example.

Actions speak louder than words. Look at the voting record and the lobbying take. The American political system is bipartisan at the top, the leaders just put up a front to dupe the rubes.

For example, Obama was a New Democrat neoliberal but campaigned broadly enough for progressives to think he supported them. He set that straight as soon as he was inaugurated and started making appointments. I am still reading on social media about progressives reporting how disappointed they were by him.

The parties campaign on hot button items that provide contrast, but the $ issues the vote elite interests based on the factions that support them, which differ somewhat but all elite factions contribute to both parties to cover their bases — to which DJT called attention and said he did the same thing for the same reasons.

Bob said...

Correct. Policies that get implemented represent action. Policies that result in gridlock represent political theater.

With the voting record we can see if there were bills that were passed by a majority on one side only. Then we can check if votes on similar bills were consistently opposed.