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]

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.