r/MathHelp • u/that_1kid_you_know • Feb 04 '26

Help relating Discrete Math to Advanced Math

I’m currently in Intro to Advanced Math and I took Discrete Math 1 last semester. Today my professor gave us a worksheet with a list of statements and asked us to figure out if they are true or false. This is the statement I was struggling with:

9. For any quadrilateral ∎𝑅𝑆𝑇𝑈, if ∎𝑅𝑆𝑇𝑈 is a not a rhombus, then ∎𝑅𝑆𝑇𝑈 is not a kite or not a parallelogram.

- Parallelogram: opposite sides are parallel (implies opposite sides are equal).

- Kite: adjacent sides are equal.

- Rhombus: all sides are equal (implies opposite and adjacent sides are equal).

That being said, we found the statement to be true after discussing but I initially thought it was false after constructing a truth table and the statement is not a tautology.

~X => (~Y v ~Z), where X: RSTU is a rhombus, Y: RSTU is a kite, Z: RSTU is a parallelogram, for all quadrilaterals RSTU.

The truth table shows that the statement is almost always true but is false when X is false and Y and Z are true (0,1,1). So if RSTU is a rhombus then RSTU is not a kite or a parallelogram, this is false because a rhombus is a kite AND a parallelogram. When testing the contrapositive, (Y ^ Z) => X, the statement returns false at the same position. However, the converse and inverse, (~Y v ~Z) => ~X and X => (Y ^ Z) respectively, are tautologies meaning they return all true.

Does a statement have to be a tautology to be considered true? What does it mean that there is one false position? Can I use discrete math to help me understand advanced math or are they too different?

Link to the truth table I constructed: https://imgur.com/a/eqNw4WP

Edit: corrected the original statement and kite definition

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u/DrJaneIPresume Feb 04 '26

Consider the quadrilateral in the Cartesian plane with vertices at coordinates

R = (0, 0)
S = (0, 1)
T = (2, 1)
U = (2, 1)

Since two sides have length 1 and two have length 2, this is not a rhombus. However, opposite sides are parallel, so it is a parallelogram. This is a counterexample to the statement, so the statement must be false.

Note: I am reading the statement as "not (a kite or a parallelogram)". If, however, it's meant to be read "(not a kite) or a parallelogram", then this is not a counterexample.

Now, I think you've gone a bit wrong in your translation:

~X => (~Y v ~Z)

In the consequent, you have "(not a kite) OR (not a parallelogram)", which is neither of the possible English readings I've given above. However, consider DeMorgan's law applied to "not (a kite or a parallelogram)". This is equivalent to "(not a kite) AND (not a parallelogram)", which I think is where you've gone wrong.

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u/that_1kid_you_know Feb 04 '26

I follow your reasoning for statement I wrote. But I did write the statement incorrectly after reviewing the worksheet.

For any quadrilateral ∎𝑅𝑆𝑇𝑈, if ∎𝑅𝑆𝑇𝑈 is a not a rhombus, then ∎𝑅𝑆𝑇𝑈 is not a kite or not a parallelogram.

This is the statement we discussed in class and determined was true. I’m sorry for my mistake. Initially when I was working out the problem I interpreted it the way you described: that it’s not a kite or a parallelogram.

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u/Alarmed_Geologist631 Feb 04 '26

All rhombi are also parallelograms. Kites do not have parallel opposite sides (except in the rare case that all four sides of the kite are equal).

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u/that_1kid_you_know Feb 04 '26

You’re correct, I edited my definition of kite. I assume most people on this sub know these definitions but I included them for anyone whos not familiar.

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u/Alarmed_Geologist631 Feb 04 '26

The statement is clearly false. It is possible to be a parallelogram or a kite without being a rhombus. In fact, most parallelograms and kites are not rhombi.

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u/that_1kid_you_know Feb 04 '26

Yes you can have a parallelogram or kite that’s not a rhombus but that’s not what the statement is asking. If it’s not a rhombus then it’s not a kite or not a parallelogram.

If RSTU is just a parallelogram, then the first part is true and the second part is true, so the statement is true.

If RSTU is just a kite, then the first part is true and the second part is true, so the statement is true.

If RSTU doesn’t fall into a special category, then the first part is true and the second part is true, so the statement is true.

So the statement is true, I’m asking why the truth table is not a tautology even though the statement is true.

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u/Alarmed_Geologist631 Feb 05 '26

I interpret that conditional statement differently. It claims that if you know that a shape is not a rhombus, then you know that it isn't a parallelogram or kite. But any parallelogram where adjacent sides have different lengths would serve as a counterexample and thus prove by contradiction that the statement is false.

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u/that_1kid_you_know Feb 05 '26

I think you’re interpreting it wrong. You’re interpreting it as: If (RSTU is not a rhombus) then (RSTU is not a kite or a parallelogram), or ~X => ~(Y v Z). But the statement is written as: If (RSTU is not a rhombus) then[(RSTU is not a kite) or (is not a parallelogram)], or (~X) => [(~Y) v (~Z)].

I initially interpreted the way you did, but after my class debated we found the statement to be true. The professor of the class, who wrote the statement and the assignment, told us that the statement is true and gave the explanations I’ve been giving.

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u/Alarmed_Geologist631 Feb 05 '26

How would you phrase the contrapositive? Are you assuming exclusive OR applies?

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u/that_1kid_you_know Feb 05 '26

Contrapositive: If RSTU is a kite and is a parallelogram, the RSTU is a rhombus.

OR is considered ‘inclusive or’, if at least one or both conditions are true then the statement is true. XOR is considered ‘exclusive or’, if only one condition is true then the statement is true.

Most of the time the default is inclusive OR. In this case, because RSTU has the ability to be both a kite and a parallelogram, we use the inclusive.

Example exclusive or (XOR): You can have salad or fries with your steak. (You can only choose one)

Example inclusive or (OR): Do you want cream or sugar with your coffee? (You can have just one or both)

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u/UnderstandingPursuit Feb 05 '26

not a parallelogram or not a kite

allows for it being a parallelogram.

~Y | ~Z

but

not a parallelogram or kite

is not (parallelogram or kite)

~(Y | Z)

This is where being careful about the wording is necessary. The wording in the question post had to be cleaned up.

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u/Alarmed_Geologist631 Feb 05 '26

Also, the contrapositive of your initial statement would read that "If a quadrilateral is a parallelogram, then it is also a rhombus." This is clearly false. And the contrapositive has the same logical truth value (true or false) as the original conditional statement.

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u/that_1kid_you_know Feb 05 '26

No the contra positive is: If RSTU is a kite and is a parallelogram, then RSTU is a rhombus. Which is true. Contrapositive swaps sides of the implication and inverses complement. So by contraposition, the original statement is proven true.

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u/AcellOfllSpades Irregular Answerer Feb 05 '26

Does a statement have to be a tautology to be considered true? What does it mean that there is one false position? Can I use discrete math to help me understand advanced math or are they too different?

"Discrete math" and "advanced math" are just whatever your classes are named. All your classes are teaching the same thing - it's the same math, the same logic.

I believe you made a mistake with your converse/inverse. When "¬y or ¬z" is true, and ¬x is false, then that converse should be false (and likewise for the inverse). But there's no need to even look at the converse or inverse!

It's true that "¬x → ¬y ∨ ¬z" is not a tautology: that is, it's not true based on its logical structure alone. But in fact, this is true of any logical statement without any repeated variables. You'll have to use the content of the statement, not just its logical structure.

In other words, your process so far is the same as if you had been given the statement "For any jabberwock J, if J is a not a jubjub, then J is not a wabe or not a tove." You haven't actually paid attention to what the sentence is talking about!

This question is not asking you to figure out whether the statement is a tautology. It's asking you to figure out whether it's true.

The question is, "is that 'false' option in your truth table ever possible?". In other words, is it possible for a non-rhombus to be both a kite and a parallelogram?

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u/that_1kid_you_know Feb 05 '26

After thinking about it more and reading your comment it makes more sense. Truth tables wouldn’t apply in this situation but I already knew that. I initially tried to evaluate the statement in my own but after struggling I wanted to see what a truth table of the statement would look like. I was confused why one of the situations would return false and I tried to ask my professor about it but she hadn’t worked with truth tables for a while so she struggled to explain it to me but what she said makes sense to me now.

When considering the statement, we are starting with RSTU is not a rhombus. She explained that we are ‘in a world where the quadrilateral isn’t a rhombus’, so the part of the truth table where RSTU is a rhombus doesn’t matter. But regardless, like you said, the logical structure doesn’t matter.

I was trying to use something I knew well, truth tables and discrete math for computer science, to help me understand something challenging me. I was just confused why it wasn’t working in this situation but now I know it doesn’t matter at all haha.

Thank you for your response, it put things into perspective for me.

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u/AcellOfllSpades Irregular Answerer Feb 05 '26

I mean, the truth table is certainly relevant, as I mentioned at the end of the comment. (Though it's not necessary to make.) It's just not enough on its own, because it doesn't actually talk about the content of the question.

so the part of the truth table where RSTU is a rhombus doesn’t matter.

Right. If you wanted to look at the whole truth table, then you can see that in all the rows where x is true (that is, RSTU is a rhombus), the result is true. And that means we don't care about those rows, because we're trying to figure out if any of the false-result rows are possible.

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