All conditions are true for a certain range

[Hypercubes]

How do you specify, in mathematical notation, that a condition is true for all cases?

For example, I want to say that $$t=5$$ for every value of $$n$$. Basically like $$\lim_{n\to\infty}{t=5}$$, but instead for every value of $$n$$, not just as $$n$$ approaches infinity. Also, how to specify the range of $$n$$ values for which the condition is true?

Thanks in advance.

 

[Char. Limit]

Well, we have the symbol ∀, which means "for all".

So you would say...

t=5 ∀n

 

[HallsofIvy]

I assume that is an example- it doesn't make sense to say that a statement, that does not involve n, is "true for all n".

But just "for all n, $n^2\ge 0$" will work or, as Char limit said, ∀n, $n^2\ge 0$.

 

[Hypercubes]

Yeah, it was just an example.

Thanks for the help!

 

[CellCoree]

To specify that a condition is true for all cases, we use the universal quantifier symbol (∀). In mathematical notation, we would write this as:

∀n, t=5. This statement reads as "for all values of n, t is equal to 5." This notation indicates that the condition is true for every possible value of n.

To specify a range of n values for which the condition is true, we can use interval notation. For example, if we want to say that the condition is true for all values of n between 1 and 10, we would write it as:

∀n∈[1,10], t=5. This statement reads as "for all values of n in the interval from 1 to 10, t is equal to 5."

Alternatively, we can also use set notation to specify a range of values. For example, if we want to say that the condition is true for all even values of n, we would write it as:

∀n∈{2,4,6,8,...}, t=5. This statement reads as "for all values of n in the set of even numbers, t is equal to 5."

Overall, the use of universal quantifiers and appropriate notation allows us to clearly specify that a condition is true for all cases and within a specific range of values.