Brackets in a disjunction

Question

How can one intuitively see that in a disjunction, putting brackets makes no difference? I mean for example: why is "At least one of the statements A, B, C, D, E, F, G is true" equivalent to "((A or B) or C) or (D or E) or (F or G)"?

Answer 1

Can you see that $\bbox[pink, 1pt]{(A\text{ or } B)\text{ or }C}$ is equivalent to $\bbox[pink, 1pt]{A\text{ or } (B\text{ or }C)}$ and therefore either can be written as $\bbox[pink, 2pt]{A\text{ or } B\text{ or }C}$ without ambiguity?

Assume the truth of $\bbox[pink, 1pt]{(A\text{ or } B)\text{ or }C}$.   This means that either $\bbox[pink, 2pt]{A\text{ or } B}$ or $\bbox[pink, 2pt]{C}$ is true.   This means at least one of $\bbox[pink, 2pt]{A}$, $\bbox[pink, 2pt]{B}$, or $\bbox[pink, 2pt]{C}$ is true.   This in turn means either $\bbox[pink, 2pt]{A}$ or $\bbox[pink, 2pt]{B\text{ or }C}$ is true.   Hence $\bbox[pink, 1pt]{A\text{ or }(B\text{ or }C)}$ is true.   Also vice versa.   Therefore the original, middle, and final statements are equivalent.

$$\bbox[pink, 1pt]{(A\text{ or } B)\text{ or }C} \equiv\bbox[pink, 2pt]{A\text{ or } B\text{ or }C} \equiv \bbox[pink, 1pt]{A\text{ or } (B\text{ or }C)}$$

All mean, "At least one of $\bbox[pink, 2pt]{A}, \bbox[pink, 2pt]{B},\bbox[pink, 2pt]{C}$ is true."

Does your intuition now agree that this principle of associativity can be extended to a disjunctive series of any length, such as:

$$\bbox[pink, 1pt]{(A\text{ or } B)\text{ or }(C\text{ or }D)\text{ or }(E\text{ or }F)}$$

...means: $\text{At least one of }\bbox[pink, 2pt]{A}, \bbox[pink, 2pt]{B},\bbox[pink, 2pt]{C},\bbox[pink, 2pt]{D},\bbox[pink, 2pt]{E},\text{ or }\bbox[pink, 2pt]{F}\text{ is true.}$