Question

I was thinking about whether there is a difference between the symbol 'x' and the expression/formula containing it. For example we discuss 'x' the symbol but equally I can talk about 'x' the expression, if a wff is defined as a subset of sequences of symbols, then the formula 'x' is a sequence containing one position with which the symbol 'x' is occurs. Is there a formal difference between these two? Mathematically a sequence is a function mapping elements to positions, mathematically this makes them different, however if I analyse a sequence 'x+y' I see that there is one occurence of both the symbol 'x' and sequence 'x'(sequence with only the symbol 'x' in it) however both correspond to the same position in the same sequence. I'm not sure there's enough information in the answer to warrant posting itasa site-appropriate answer proper, but here it is: In the mathematics of permutations, for a set ofnelements, you can put those elements inn!-many sequences. For example, if you have 4 elements, you can put them in 24 sequences. The cases of an empty set and a set with only one element then have relatively trivial sequences to their names, since 0! = 1 (stipulated) and 1! = 1. Pragmatically/in natural language, we tend not to have a reason to think of such trivialities as sequences, since we often speak of e.g. "a sequence of events," and it would conserve time in written or spoken word usage to refer to "an event" when we are considering only one event. So pedantically, a sequence can contain one stage, but I'm only a pedant some of the time, a peasant the rest, and in this case I'd tend to side with my fellow peasants. Yes there's a difference; the symbols and an expression consisting of symbols are two different objects with different functions that you can apply to them, producing different results. A common example that's widely seen in computer programming is the difference between alistof objects and the objects from which we compose lists, which I will here callatoms.If I declare my set of atoms to be letters and numbers I can make lists such as(x 1 y 2),(x)and(2), and apply functions such asheadandtailto these: head (x 1 y 2) ⇒ x

tail (x 1 y 2) ⇒ (1 y 2)
head (3) ⇒ 3
tail (3) ⇒ () You cannot applyheadortailto an atom; they operate only on lists. It makes no sense to ask, "what is the head of 3" as atoms have no head or tail. Notice thatheadtakes a list and produces an atom, buttailtakes a list and produces another list. You can see from this that (at least some) functions take notanyother object as their argument, but only objects of specifictypes.By distinguishing between the symbol 'x' and the expression consisting of only the symbol 'x' you have created two types in your language, making them two different objects. A. Mathematical Sequences
Formal Symbols vs. Expressions B.Permutations and Sequences C. Objects and Functions in D. Computer Programming

Answer 1

The symbol 'x' and the expression containing it have different mathematical purposes and interpretations. The symbol 'x' represents a variable or unknown quantity, while the expression 'x' refers to a specific sequence with the symbol 'x' at a particular position. Understanding their differences is important in mathematics.

Step-by-step explanation:

The symbol 'x' and the expression containing it serve different mathematical purposes and have different interpretations.

The symbol 'x' is typically used to represent a variable or unknown quantity in an equation or formula. It is a placeholder that can take on different values.

On the other hand, the expression 'x' refers to a specific sequence or collection of symbols, where the symbol 'x' appears at a particular position. While both the symbol and the expression share the same symbol 'x', they have distinct mathematical properties and functions.

For example, in the equation 2x - 5 = 10, the symbol 'x' represents an unknown value that we are trying to solve for. It can take on different values, such as 3 or -2, depending on the solution. On the other hand, the expression 'x' represents a sequence with the symbol 'x' occurring at a specific position. In this case, the expression 'x' contains only one position with the symbol 'x'.

Understanding the difference between the symbol 'x' and the expression 'x' is important in mathematics as it helps clarify the context and purpose of their usage in equations, formulas, and mathematical representations.