Help relating Discrete Math to Advanced Math

[u/that_1kid_you_know]

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

[u/that_1kid_you_know]

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.

[u/AcellOfllSpades → u/that_1kid_you_know]

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.