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It is from maths workshop course, we were asked to prove statement below.

The set $A=\{a,b,c\}$ is a finite set containing three elements.

I rewrote this as:

Let A={a,b,c} be a set $\iff$ A is a finite set conaining three elements.$\qquad (\times)$

And began to prove it as implication in two ways.

The lecturer said that my re-statement is incorrect, but that it should be instead:

Let A be a set containing three elements: a,b,c. Then A is finite set with three elements.

or equivalently:

$A=\{a,b,c\}$ is a set $\implies A$ is finite set with three elements.

I wasn't given reason why my restatement $(\times)$ is incorrect. Is lecturer correct by calling it incorrect, could anyone comment on this?

flowian
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2 Answers2

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I would say it is because you only stated that $A$ has three elements and these three elements could be anything and not only $\{a,b,c\}$.

So if you assume that A has three elements you cannot conclude that the three elements are $a,b$ and $c$.

However if you concluded that there exists a set $A=\{a,b,c\}$ then it would be more correct.

user866453
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First of all, it should be noted that $a,b,c$ are distinct, otherwise $A = \{a,b,c\}$ does not contain three elements.

Moreover, the problem in your restatement seems to lie in the direction $\Leftarrow$: it lacks quantifiers. One should write

$$A \text{ is a finite set with three elements} \implies \exists \text{ distinct elements } a,b,c \text{ such that } A = \{a,b,c\}.$$

In other words, you make it seem that $a,b,c$ are fixed from the beginning. If they are, then $\Leftarrow$ is incorrect. If they are not, well, then you'll need to quantify them.

Marktmeister
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