Equation in discrete mathematics

Question

I had something task on discrete mathematics, which I couldn't resolve.

Is

$(3,3) = (3,3,3)$

True or False? And why?

Any help will be highly appreciated :)

Answer 1

There are three kinds of collections-of-objects that you might distinguish between:

• Sets: unordered collections without multiplicity. Here, $\{3,3\}$ and $\{3,3,3\}$ are both equal and the same set as $\{3\}$; $\{1,2,3\}$ and $\{3,2,1\}$ are also equal.
• Multisets: unordered collections with multiplicity. Here, $[3,3]$ and $[3,3,3]$ are different, but $[1,2,3]$ and $[3,2,1]$ are equal. (The bracket notation is less standard than the two others; some people even use the same notation as for sets.)
• Tuples or sequences: ordered collections with multiplicity. Here, $(3,3)$ and $(3,3,3)$ are different, and so are $(1,2,3)$ and $(3,2,1)$.

In most cases, you can conclude from $\{$ and $\}$ braces that we're talking about sets, and from $($ and $)$ braces that we're talking about tuples. But all notation is convention, and so you might have other information that contradicts this.

(For example, in the context of abstract algebra, $(3,3)$ and $(3,3,3)$ might both denote $(3)$, the ideal generated by $3$. In this case, the generators are a set, not a sequence, despite the notation.)

Answer 2

What do you mean by these symbols? For example, let us assume that $(3,3)$ and $(3,3,3)$ refers to ordered pair/tripple of numbers. Then the element $(3,3)$ lives in the set $\mathbb{N} \times \mathbb{N}$ but $(3,3,3)$ lives in $\mathbb{N} \times \mathbb{N} \times \mathbb{N}$ thus they are (quite essentialy) two completely different objects as they do not even share the common realm of existence, so to speak :D. On the other hand, if $(3,3)$ is a notation (and not convenient one) of a set of two elements, i.e. would be classicaly written in the form $\{ a,b \}$, then $(3,3,3)$ and $(3,3)$ is the same thing because it is actually a set containing only one element, namely the symbol $3$. But this second case assumes quite a bad notational convention so the first case is more probable, thus I would say the answer to your question is False.