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In the Jan SAT (math section 3, question 15, for those of you who have received the booklet in the mail), there is a symbol which is a shaded-in triangle pointing downward. What does this symbol mean in general, and can you explain this SAT comparison question:
(I will use the symbol "$" since I can't type that kind of triangle.)
Let x$y be defined for all integers x and y by
x$y=xy-(x+y)
Comparison choices:
(1$2)$3 . . . . . 1$(2$3)
Answers:
A if the quntity in Column A is greater
B if the quantity in Column B is greater
C if the two quantities are equal
D if the relationship cannot be determined from the information given.

By Timepass (Timepass) on Monday, March 10, 2003 - 08:03 am: Edit

The answer is B

By Texas137 (Texas137) on Monday, March 10, 2003 - 10:04 am: Edit

That's a "funny symbol problem". There's at least one on almost every SAT. It is not a standard math symbol.It can be anything - a star, a smiley face, an @, or the dollar sign you're using as a substitute. Basically it's a "trick". They're inviting you to freak out about the symbol and give up on the problem. But the problem defines what they want you to do when you see the symbol. In your case, whenever you see the "funny symbol" btwn. 2 numbers you're supposed to multiply them together then subtract the sum. So...
(1$2) = 2-3 = -1
(-1)$3 = -3-2 = -5 for column A
(2$3) = 6-5 = 1
1$1 = 1-2 = -1 for column B
-1 in column B is greater than -5 in column A, so the answer is B

By Incognito (Incognito) on Thursday, March 13, 2003 - 05:57 pm: Edit

ignore the operations and symbols. Just realize that its the product minus the sum. Then solve.

By Natedawg (Natedawg) on Thursday, March 20, 2003 - 11:40 pm: Edit

LOL... As others have said, this is NOT a math question. Any question which employs an un-defined symbol is considered (by mathematicians in particular) to be a Meaningless question! These 'made-up' symbols have no operational definition, so they mean exactly NOTHING. BTW, when you get to college don't even think about using 'made-up' math symbols on any math assignment or test, you will get marked down for this. -Nathan

By Dferraro (Dferraro) on Friday, March 21, 2003 - 12:21 am: Edit

They actually do mean something...it is a FUNCTION, probably the MOST IMPORTANT thing in mathematics..the behavior of functions. you could easily write this as f(x)=xy-(x+y)..........

By Natedawg (Natedawg) on Friday, March 21, 2003 - 02:48 am: Edit

Well, obviously the testee is supposed to interpit the random symbol as functional notation, and of course functions are the bedrock of mathematics. However, in the problems we are discussing the testee is supposed to 'assume' the meaning of the random symbol, as this is the only way to get the right answer. However, this doesn't mean the question is a math question! A problem that uses un-defined symbols cannot be considered a valid math problem. If the symbol(s) used have no standard definition, then some type of citation MUST be used to properly define the symbol. So when you said f(x)=xy-(x+y)... "could just as easily have been used", I will have to disagree with this.


Consider: The symbol f(x) indicates a functional relationship exists, with respect to the problem under consideration (namely the variable x). The operators 'f', '()' and 'x' all have definitions in mathematics. On the particular SAT problems in question, treating the random symbols as a functional notation is the fairly obvious way to achieve the 'correct' answer. However, I could easily construct problems which employ a random symbol, that 'appear' to indicate a simple substitution of quantities, but the only way to achieve the 'correct' answer would be to discover MY definition of what the symbol meant. What if my symbol, say '*', indicated the independant variable be raised to the (1/343) power, then multiplied by the complex number 6i , in order to produce the 'correct' answer?? It would at first glance 'appear' the symbol '*' indicated functional notation, but in order to produce the correct answer, the testee would have to 'back out' my definition by 'logic'.

The point here is, in both problems un-defined symbols are used. In my problem, formulating the 'meaning' of the symbol from logical induction would be near-impossible. In the SAT problem, it is much easier. But the fact that making this assumption is 'easier' to do on the SAT problem than in my problem says NOTHING about whether either question is a valid mathematical question. -Nathan

By Dferraro (Dferraro) on Friday, March 21, 2003 - 02:51 am: Edit

wtf-- i thought i edited my message to say:

f(x,y)=xy-(x+y)

By Brd (Brd) on Saturday, March 22, 2003 - 01:10 pm: Edit

"These 'made-up' symbols have no operational definition, so they mean exactly NOTHING."

Nate, I really have to disagree with this. The problem itself gives a _definition_ for the binary relation:

"Let x$y be defined for all integers x and y by x$y=xy-(x+y)"

Working mathematicians invent and define new notation on the spot perhaps more often than you would imagine. When people are creating new mathematics there is no 'standard referecnce' to cite, after all.

By Xiggi (Xiggi) on Saturday, March 22, 2003 - 02:40 pm: Edit

See my other post with this strange sign: ñ


W + B = 17
B ñ W = 22

Column A -4W
Column B 10

A. Column A is larger
B.if Column B is larger
C.if the columns are equal
D.if there is not enough information to decide

By Natedawg (Natedawg) on Tuesday, March 25, 2003 - 01:28 pm: Edit

"The problem itself gives a _definition_ for the binary relation: ... Working mathematicians invent and define new notation on the spot perhaps more often than you would imagine. When people are creating new mathematics there is no 'standard referecnce' to cite, after all."

I'm sorry, I still disagree with you on this. To begin with, when you said 'binary relation', were you referring to the whole equation, or too the symbol "$"? How do you know what type of relation "$" indicates? No footnote or definition is provided for the symbol "$", so how do you know what kind of symbol it is? Of course mathematicians create new or un-conventional notations, symbols etc... when it's helpful. But when new or un-conventional notation is introduced, a citation or footnote is always provided to explain what the notatation means, in that particular context. I referred to "$" as a symbol, because this is ALL you know about it. Is this symbol an operator, such as multiplication, or perhaps a variable? Or is it a constant? You don't know, so you have to GUESS by seeing what values fufill the given equation. You could argue

"well, if you just look at the problem, it is
obvious the symbol is a multiplication
operator, because this is the only choice that
would line up with the context of the problem"

But is this really true?? Our problem was


"Let x$y be defined for all integers x and y by x$y=xy-(x+y)


We may now consider some possible cases, whereby a particular definition for the symbol "$" would satisfy our given operations:

Case (1): the symbol "$" could stand for a multiplicative constant a , where x and y are integers. (BTW, I will use * to indicate multiplication)

$=some constant a. so we have

x$y=xy-(x+y) =

x*a*y=xy-(x+y) =

axy=xy-(x+y), where x and y are integers.

In the statements (1$2)$3 and 1$(2$3), if "$" is some constant a, we can write these as:

(1$2)$3= (1*a*2)*a*3 = (2a)*3a =6a^2

1$(2$3)=1*a*(2*a*3)=a(6a)= 6a^2

6a^2=6a^2

In this case, our answer would be "C, the two quantities are equal".

The assumption that "$" is some multiplicative constant a appears to be just as 'valid' as any other assumption one could make. I could give more cases which produce other 'correct' answers, but I don't want to spend the time typing into this tiny box. x and y are limited to integers, but the symbol "$" has no such limitations. By assuming that "$" is a constant multiplied by the values x and y, an entirely different set of results is produced. The question ultimately becomes "What do I assume"? If the problem involves trying to figure out what the test-writers would assume, and your not allowed to ask, the problem leaves the realm of "mathematics". -Nathan

By Frick (Frick) on Tuesday, March 25, 2003 - 01:58 pm: Edit

an upside-down triangle is 'del', or the gradient operator, and indicates the partial derivative of a function... but it's not usually shaded in

By Brd (Brd) on Tuesday, March 25, 2003 - 02:12 pm: Edit

The actual technical name for a (hollow) upside down triangle symbol is 'nabla' although colloquially it is often referred to as "del". It can be used to denote the gradient (and when it does, it is never shaded in) but is also used in notation for the divergence ("del-dot")and curl ("del-times") operators.

By Brd (Brd) on Tuesday, March 25, 2003 - 02:27 pm: Edit

Oh Nate, my apologies, I had not seen your response. I have to say I still don't agree. While I suppose it would have been a courtesy for them to explicitly mention that "$" denotes an infix binary relation, I think it is clear enough from context. In any event, your example is not valid. The problem clearly states that

x$y is _defined_ for all integers x, y by x$y=xy-(x+y)

This is what is called an operational definition. The important point to note is that the relation must hold for *all* x and y. That is why the right hand side can be considered to define the left hand side. In your example

x*a*y = xy - (x+y)

For any particular a, the relation is patently false for almost all values of x and y. "$" cannot be what you suggest it might be here.

But the main point remains, you say that "$" needs to be defined by some footnote or citation. I claim the operational definition stated in the problem *is* that very definition, and that you are trying to make an issue out of nothing. This kind of operational definition is how mathematicians and physicists present things like differential operators all the time.

By Natedawg (Natedawg) on Tuesday, March 25, 2003 - 04:27 pm: Edit

My mistake, a is false in most cases. Still, I am not entirely certain the statement

x$y is _defined_ for all integers x, y by x$y=xy-(x+y)

is a proper operational definition with $ being a multiplication operator; Especially considering the aforementioned fact that the actual symbol employed (nabla) commonly denotes partial derviatives of a function. Beyond that, I am not sure if this statement qualifies as a binary relation. I am not terribly familiar with the Set theory involved, but I will read up on it in the next few days. -Nathan