# What's the meaning of $\succ$ operator?

What is the meaning of $$\succ$$ symbol? I have an snippet which includes this operator: (Article is about choice theory)

"a complete and transitive preference $$\succ$$ over X"

Note that $$\succ$$ is Unicode character U+227B.

• The same thing worded differently: Let $\succ$ be a complete and transitive preference over $X$. Jan 20, 2014 at 14:12
• This symbol also has a different meaning in mathematical economics. The symbol $≻$ can be that given A $≻$ B, that the choice A is preferred over B. For example, becoming a doctor might be preferred over becoming a social worker, but it really would not make sense to say being a doctor is greater than being a social worker. May 23 at 14:10

$\succ$ is named "succeeds" and is the name given by the author to the complete and transitive preference (~order). You can read it as a $>$ for simplification. Other uses of the symbol occur for example with matrices. $A\succ 0$ means that $A$ is a positive definite matrix (i.e. $x^T A x > 0 \quad\forall\ x\neq 0$)

• @SuatAtan happy to help. If this solved your question, please mark it as answered by clicking on the checkmark to the left of the answer. If not, feel free to ask for further clarification. Jan 22, 2014 at 18:05
• Unfortunately my reputation is not enough (<15) for voting. But after attaining this value i'll. Feb 21, 2014 at 9:10
• @SuatAtan Accepting answers doesn't fall within this restriction, see here for a description. Feb 23, 2014 at 13:25

Here is how you would find an explanation on the internet.

Entering the keywords "complete transitive preference choice theory," the first hit is this wikipedia page on preference. In the section basic premises, they start to define symbols for preference, and then in this later section they define what a complete and transitive preference order is.

I have a slight feeling you might be too focused on the symbol $\succ$ itself. The wording suggests you might believe that the symbol has one and only one meaning. This isn't a very practical viewpoint since many symbols have uses in different concepts, and sometimes the same concept uses multiple symbols. (In case my feeling completely wrong, I apologize in advance!)

Aside from the symbol $\succ$, an author could choose to use $>$ or anything else shaped like that. So the phrase

"a complete and transitive preference $\succ$ over X"

already completely explains what $\succ$ is: it is a complete and transitive preference over $X$. If one is still puzzled at this point, then the right question is what is a complete and transitive preference order? not "What's this symbol?" The symbol isn't that important, but what it denotes is important.

• Or if you see preference as something negative, you could chose $\prec$ or $<$; in fact about anything that has a "direction". Look over here for some examples. Jan 20, 2014 at 15:09
• Thanks rschwieb for your answer. Frankly, your advise about interpreting statements which have math symbols, is instructive for me. But as being newbie, suffocating in abstract terminology and sentences sometimes causes kind of obssession and misfocusing. So perhaps, this web app is great and like lighthouse for this reason. Thanks again. Jan 21, 2014 at 8:10
• @SuatAtan Awesome, then my gamble paid off! Don't worry, I think a lot of students hit the same snag. But I'm sure you'll rapidly get the hang of this thought pattern with practice. Jan 21, 2014 at 11:14

Note that $$(\cdots 6 > 5 > 4 > 3 > 2 > 1 > 0 > \cdots)$$.

The succeeds operator $$(≻)$$ is a generalization of idea behind some numbers being bigger than others.

For example, given any two plant-based edible foods $$f_{1}$$ and $$f_{2}$$ we let $$f_{1} ≻ f_{2}$$ if and only if $$f_{1}$$ is more like a fruit than $$f_{2}$$.

For example, you might have the following:

The set of very good-tasting fruits:

$$\{ \mathtt{Apple}, \mathtt{Banana}, \mathtt{Blackberry}, \mathtt{Blueberry}, \mathtt{Cantaloupe}, \mathtt{Cherry}, \mathtt{Grape}, \mathtt{Guava}, \mathtt{Lemon}, \mathtt{Lime}, \mathtt{Mangoes}, \mathtt{Orange}, \mathtt{Peach}, \mathtt{Pears}, \mathtt{Pineapple}, \mathtt{Plums}, \mathtt{Raspberry}, \mathtt{Strawberry}, \mathtt{Watermelon} \}$$

The set of fruits which do not taste as good as a strawberry. However, some American school children pride themselves on telling people that one and only one element of this set is a fruit:

$$\{\mathtt{tomato} \}$$

Plants have leaves, stems, roots, flowers, fruits, and seeds. In the botanical sense, all of the following are fruits:

$$\{ \mathtt{bell\ pepper}, \mathtt{courgette\ zucchini}, \mathtt{cucumber}, \}$$

Finally, we have five more of the worlds many edible plants. For these five, people eat the stems, leaves, seeds, or roots more often than the fruits.

$$\{ \mathtt{broccoli}, \mathtt{carrot}, \mathtt{lettuce}, \mathtt{onion}, \mathtt{potato} \}$$

And so, we have:

$$\mathtt{Watermelon} ≻ \mathtt{Tomato} ≻ \mathtt{Cucumber} ≻ \mathtt{Cucumber} \mathtt{Broccoli}$$

Given any two plant-based edible foods $$f_{1}$$ and $$f_{2}$$ we say $$f_{1} ≻ f_{2}$$ if and only if $$f_{1}$$ is more like a fruit than $$f_{2}$$.

Relatively speaking, a model of the world which uses various shades of grey, is closer to reality than a world which is pure black and white.

Let $$\mathcal{BLACK}$$ be the following set:

$$\{ \mathtt{Apple}, \mathtt{Avocado}, \mathtt{Banana}, \mathtt{Blackberry}, \mathtt{Blueberry}, \mathtt{Cantaloupe}, \mathtt{Cherry}, \mathtt{Grape}, \mathtt{Guava}, \mathtt{Lemon}, \mathtt{Lime}, \mathtt{Mangoes}, \mathtt{Orange}, \mathtt{Peach}, \mathtt{Pears}, \mathtt{Pineapple}, \mathtt{Plums}, \mathtt{Raspberry}, \mathtt{Strawberry}, \mathtt{Watermelon }\}$$

Let $$\mathcal{CHARCOAL\_GREY}$$ be the following set:

$$\{\mathtt{tomato} \}$$

Let $$\mathcal{GREY}$$ be the following set:

$$\{ \mathtt{bell pepper}, \mathtt{courgette\ zucchini}, \mathtt{cucumber} \}$$

Let $$\mathcal{WHITE}$$ be the following set:

$$\{ \mathtt{broccoli}, \mathtt{carrot}, \mathtt{lettuce}, \mathtt{onion}, \mathtt{potato} \}$$

# What about $$2 > 1$$ ?

Once upon a time, a German person used the symbol $$ℤ$$ to represent the set of all numbers like $$-71, -18, 0, +1 +2$$".

To digress, the Germans call themselves "Deutsch", more often than they call themselves "Germans".

The zalen do not include decimal numbers like $$2.5$$ or $$\pi = 3.14 \dots$$

Ever since a Deutsch man used the symbol $$ℤ$$, American mathematicians have used the letter $$ℤ$$.

The zahlen are the whole numbers both negative and positive.

The zalen are normally put into a chain.

For any $$x$$ element of $$ℤ$$ $$f(x)$$ is just $$x + 1$$.

You can get $$f^{n}(x)$$ by adding $$1$$ many times.

• $$f^{0}(x) = x$$
• $$f^{1}(x) = x+1$$
• $$f^{2}(x) = (x+1)+1$$
• $$f^{3}(x) = ((x+1)+1)+1$$

For negative numbers, we have stuff like $$f^{-5}(x) = x - 5$$

We have for any $$n \in ℤ$$, $$(f^{n} x) = (f^{n-1} x) + 1$$

The rules above about adding one a bunch of times to a zahlen are inspired by the peano axioms.

Note that $$f^{n}(x)$$ means the same thing as $$(f^{n} x)$$.

Normally, two numbers are separated by an infix operator or delimiter. For example we have a + between 1 and 2 in the expression 1 + 2.

In the case of $$f^{n}(x)$$, the infix delimiter is a transition from alphanumeric characters to non-alphanumeric characters.

The parentheses () simply show us which inputs should be grouped together before function f is applied to those inputs.

The greater-than-sign $$>$$ defined as follows for the zählen numbers (for zero, the whole numbers both positive and negative).

for any $$x$$ element of $$\mathbb{Z}$$ and for any $$n$$ element of $$\mathbb{Z}$$, if $$n > 0$$, we have $$f^{n}(x) > x$$.

If you add the number $$1$$ over and over again to a number $$x$$, then that result will be greater than number $$x$$.

The succeeds operator $$(≻)$$ is like $$10 > 4$$, except that it applies to objects other than numbers.