Counterexamples??

New member
it says state a counterexample to the following definition:
a circle is a figure that is round.
A. A square is a figure
b. some ellipses are small
c. a circle can be big
d. ovals or ellipses are also round

I chose a, but Im sure if its right, or how youre able to tell if its a counterexample.

tkhunny

Moderator
Staff member
a) Refuting only half the premise. Squares are not round in any sense of the word.
b) Why do we care about size?
c) Size again. Who cares?
d) Closer, but did we establish that ellipses are figures?

It's a pretty weak question and answer, but I'd go with d.

lookagain

Senior Member
it says state a counterexample to the following definition:

A circle is a figure that is round.

a. A square is a figure.
b. Some ellipses are small.
c. A circle can be big.
d. Ovals or ellipses are also round.

I chose a, but Im sure if its right, or how youre able to tell if its a counterexample.
None of the choices are correct. The words "small" and "big" are subjective in meaning.
Being "round" does include allowing figures to not be circles or to not be spheres, for example.

The definition and choice "d" are true as is, so "d" is not a counterexample.

The given definition does not state the equivalent to "If a figure is round,
then it is a circle." And it should. That is an important distinction.

The last choice should be given as "Ovals $$\displaystyle \text{and} \ \$$ ellipses."

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The whole problem should be closer in meaning to the following lines,
if it is to make sense and have a correct choice:

State a counterexample to the following definition:

$$\displaystyle If \ a \ figure \ is \ round, \ then \ it \ is \ a \ circle.$$

$$\displaystyle a. \ \ A \ square \ is \ a \ figure.$$

$$\displaystyle b. \ \ Some \ ellipses, (and \ all \ ellipses \ are \ figures), \ are \$$ ____________ .

$$\displaystyle c. \ \ A \ circle \ can \ be \$$ _____________ .

$$\displaystyle d. \ \ Ovals \ and \ ellipses \ are \ also \ round.$$

If it were the problem immediately above (with appropriate words in the blanks),
then choice "d" would be correct.