Do different letters (signifying digits) always mean different digits?

Question

For problems like the below, do we always assume that the four integers will signify different numbers? Because otherwise, if w can be equal to x, the answer will be different.

1. If set S consists of the positive integers w, x, y, and z, is the range of the numbers in S greater than 6 ?
(1) No two numbers in set S are consecutive.
(2) None of the numbers in set S are multiples of 3.

For the above example, while the correct answer is C, if any of the four digits w, x, y and z could be equal to each other, like 2, 2, 4, 7 for example, then the answer would be E. So how do we know for sure when the question does not specify that the integers are all different? Do we assume that because of the "different letters" used?

Another example of such a problem where it is vague -

2. If x = 0.abcd, where a, b, c, and d each represent a nonzero digit of x, what is the value of x ?
(1) a/2 = b = 2c = 4d
(2) The product of a and d is equal to the product of b and c.

In the above example, if we only looked at option 2, could we consider that all four digits could be equal to 1? Or the different letters a, b, c, d automatically mean different digits?

nick1816 · Answer

If different letters are assigned to the variables, it doesn't mean that they can't take same values.Hence, a,b,c and d can have same values in Q2. But in Q1, variables w, x, y and z must be distinct because  a set  always contains  distinct  elements.

Bibah · Answer

Thank you so much. Question 2 confused me because the explanation I received from GMAT Hacks stated-
"Statement (2) is insufficient: a, b, c, and d could be 1, 2, 4, and 8 or 1, 2, 3, and 6."

It didn't use repetitive numbers as an example so I was wondering if that is not an option.

IanStewart · Answer

In general, in math, if you see two unknowns x and y in a problem, then x and y can be equal, unless the question tells you otherwise. But the word "set" in math has a precise definition -- in a "set", all the values must be different. But because the question above is so carelessly worded (it's clearly not an official problem), I wouldn't trust the question writer to understand the definition of a "set", so I wouldn't know here what we can and cannot assume about the unknowns. On the real GMAT, they rarely use the word "set" in statistics questions -- they normally talk about "lists" or "data sets", and in a list or data set, you are allowed to have identical values.

You could never see a statement on the GMAT that reads like Statement 1 here. "No two numbers are consecutive" does not mean anything unless you specify what kind of sequencing your discussing -- "consecutive integers" or "consecutive multiples of 3" or "consecutive prime numbers"? I assume it means to say they are not consecutive integers, but that's just a guess.

Any time you're spending here trying to decode the language is time wasted, since you'll never need to worry about ambiguous wording on the real GMAT. So I'd suggest studying from sources with properly constructed problems (official sources, for example).

Do different letters (signifying digits) always mean different digits?

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