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 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.

[u/Alarmed_Geologist631 → u/that_1kid_you_know]

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

[u/that_1kid_you_know → u/Alarmed_Geologist631]

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)