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∀n : ((0≤n<19) →(A(n) < A(19))

I want to know what this statement means. I'm new to Logic and I'm starting to understand parts of it (mainly the basics) however I've come up stuck when it comes to this statement. A relates to an array but i can't tell whether the statement is saying there's 19 objects in the array or something else.

Also I'm confused what this statement also means.

∀n : (A(0)=2 ˄ (n>1 → A(n)=2 * (A(n-1))))

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    If $A(n)$ is an "array" of numbers, the first statement means: each one of the first $19$ elements of the array (i.e. the elemnts $A(0), A(1), \ldots A(18)$) is less than $A(19)$. – 2017-01-13
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    The secons statement means : $A(0)$ (the first element of the "array") is equal to $2$ (it makes sense: the array is a list of numbers...) and each element of the array (starting from the second) is twice the preceeding one, i.e. $A(1)=2 \times A(0)= 2 \times 2=4$, and so on... – 2017-01-13
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    May I ask why A1 is equal to 4. Doesn't this part n>1 Suggest that if n = 1 the previous value doesn't get multiplied by 2? – 2017-01-13
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    @SylvesterPeter you are right... It actually never specifies what $A(1)$ is .. Just that whatever $A(1)$ is, $A(2)$ is twice that, and $A(3)$ twice $A(2)$ etc. But as Fabio said, they probably meant $n>0$ rather than $n>1$ in that expression, so that $A(1) = 2*A(0) = 2$, $A(2) = 4$, etc. – 2017-01-13
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    thanks you've cleared this up for me – 2017-01-17

2 Answers 2

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∀n : ((0≤n<19) →(A(n) < A(19)) means that given any number n between 0 and 19 (including 0, but not including 19) it is true that A(n) < A(19). In other words, any of the entries A(0), A(1), A(2), ... A(18) of the array A are smaller than A(19).

∀n : (A(0)=2 ˄ (n>1 → A(n)=2 * (A(n-1)))) means that the 'first' entry A(0) equals 2, and that all other entries are twice the previous entry ... so A(1)=2, A(2)=4, A(3)=8, etc. .... Or at least I am pretty sure they tried to say that, but by using $n>1$ in the expression they actually don't specify what $A(1)$ is..., so it would merely say that whtever $A(1)$ is, $A(2)$ is twice that, etc. ... By changing this to $n>0$ you get what I just said.

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    Thank you very much you've helped me a lot. – 2017-01-13
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$\forall n : ((0 \leq n < 19) \rightarrow (A(n) < A(19))$ says that the first nineteen elements of the array are all strictly less than the twentieth element. You read it this way, "Whenever $n$ is a number between 0 (included) and 19 (excluded), $A(n)$ is less than $A(19)$." Nothing is said about the other elements of the array.

$\forall n : (A(0) = 2 \wedge (n > 1 \rightarrow A(n) = 2 \cdot A(n-1)))$ says that $A(0)$ equals $2$, and for $n > 1$ (this is probably a typo: it makes more sense if it is $n > 0$) $A(n)$ is twice $A(n-1)$.

Written this way, these formulae refer to infinite arrays, or if you prefer, functions that map natural numbers to numbers of some kind.

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    thank you very Fabio much this was very helpful. – 2017-01-13