False converse example?

biohaiid

I think this was already asked but can't find the thread.
Can someone help me out?
A simple enough one to remember.

MathsManiac

I assume you mean you want a statement that's true but the converse of which is false.

Simple one: if x is greater than 3, then x must be greater than 2.

(The converse of it is: "if x is greater than 2, then x must be greater than 3".which is false)

Do you want a geometry one? "If a triangle is equilateral, then it is isosceles".
"If a quadrialteral is a square, then it is a rectangle".

You can make them up easily enough, if you start thinking about different shapes and their properties.

Incompetent

What's this for?

BL1993

All squares are rectangles but not all rectangles are squares.

MathsManiac

Incompetent wrote: »

What's this for?

One of the words that you're supposed to know the meaning of is "converse". To find out whether you know what that means, on one of the papers, they asked you to write down a statement that was true but for which the converse was false. Question 6A on this paper: http://www.examinations.ie/schools/Project_Maths_Phase_2_P2_Ordinary_Level.pdf

Booom!

MathsManiac wrote: »

I assume you mean you want a statement that's true but the converse of which is false.

Simple one: if x is greater than 3, then x must be greater than 2.

(The converse of it is: "if x is greater than 2, then x must be greater than 3".which is false)

Do you want a geometry one? "If a triangle is equilateral, then it is isosceles".
"If a quadrialteral is a square, then it is a rectangle".

You can make them up easily enough, if you start thinking about different shapes and their properties.

Im sure he's looking for an example of a theorem with an untrue converse

Fergus_

MathsManiac wrote: »

One of the words that you're supposed to know the meaning of is "converse". To find out whether you know what that means, on one of the papers, they asked you to write down a statement that was true but for which the converse was false. Question 6A on this paper: http://www.examinations.ie/schools/Project_Maths_Phase_2_P2_Ordinary_Level.pdf

I thought I was reading for P2 then I saw that link

Fergus_

MathsManiac wrote: »

I assume you mean you want a statement that's true but the converse of which is false.

Simple one: if x is greater than 3, then x must be greater than 2.

(The converse of it is: "if x is greater than 2, then x must be greater than 3".which is false)

Do you want a geometry one? "If a triangle is equilateral, then it is isosceles".
"If a quadrialteral is a square, then it is a rectangle".

You can make them up easily enough, if you start thinking about different shapes and their properties.

Would you mind providing one for a "corollory"?

Would, "3 angles in a triangle = 180 degrees" be an example?

ChemHickey

Fergus_ wrote: »

Would you mind providing one for a "corollory"?

Would, "3 angles in a triangle = 180 degrees" be an example?

A corrolary would be

Theorem: The angle at the centre is twice the measure of any angle standing on the same arc

Corrolary: the angle subtended by the diameter (180) is a right angle.(90)

Eathrin

Fergus_ wrote: »

Would you mind providing one for a "corollory"?

Would, "3 angles in a triangle = 180 degrees" be an example?

No that's a theorem I believe.

Say an angle on a line is 180 deg.

Fergus_

ChemHickey wrote: »

A corrolary would be

Theorem: The angle at the centre is twice the measure of any angle standing on the same arc

Corrolary: the angle subtended by the diameter (180) is a right angle.(90)

Thank you!

ChemHickey

Eathrin wrote: »

No that's a theorem I believe.

Say an angle on a line is 180 deg.

But that would be an axiom would it not be? It's not derived from any theorem is it?

Nazata

ChemHickey wrote: »

But that would be an axiom would it not be? It's not derived from any theorem is it?

I was thinking the same thing, I'm pretty sure it is an axiom...

MathsManiac

Booom! wrote: »

Im sure he's looking for an example of a theorem with an untrue converse

Well all of those things I mentioned are theorems whose converses are false, (bearing in mind that a theorem is simply a mathematical statement than can be proven.)

If you want an actual statement of a result that's listed on the LC geometry course, and whose converse is false, see Remarks 1 and 2 on page 52 of the syllabus, along with the surrounding text: http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/LC_Maths_for_examination_in_2012.pdf. (Also, Remark 6 on page 60).

MathsManiac

ChemHickey wrote: »

But that would be an axiom would it not be? It's not derived from any theorem is it?

Yes, the statement that a straight angle is 180 degrees is an axiom. It's axiom 3 on page 40 of the syllabus: http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Project_Maths/Syllabuses_and_Assessment/LC_Maths_for_examination_in_2012.pdf

biohaiid

Booom! wrote: »

Im sure he's looking for an example of a theorem with an untrue converse

*She
Jesus, second time that's happened now. :L

Thanks for all your help, but Iv'e given up all hope of paper 2. I am prepared to now fail.

whiteandlight

You all seem to be mixing up the meanings of several terms:

Axioms:statements we accept to be true without further proof. We later use these to prove theorems.
Example:
there are 180 degrees in a straight line,
Distance cannot be negative

Theorems: statements we prove to be true (often using axioms or previously proved theorems)
Examples: the three angles in a triangle add to 180
The exterior angle of a triangle is equal to the sum of the remote interior angles

Corrollory: a statement that follows obviously from a theorem. (sort of an accidental theorem if you will. Proving the theorem automatically proves the corollary
Example: the angle standing on a diameter is 90 degrees. (this is because the diameter angle is 180 and the theorem proves the able standing on that arc must be half it ie 90 degrees)

Converse: the reverse of a statement.

It has nothing to do with any of the above except that tHe converse of theorems/axioms/corrollarys tend to be mathematically interesting. sometimes reversing a theorem or a geometry statement can give a false converse or a sentence that makes no sense, sometimes it's true. In the exam paper question they were simply testing could you construct the converse of a statement accurately.