Discrete Math for Data Science

DSCI 220, 2025 W1

September 15, 2025

Announcements

• Enroll in PrairieTest here: https://us.prairietest.com/pt/student/course/12361/enroll/392674810508
• Sign up for EX1. Practice materials on PL.
• Tutorials begin this week.

Quantifiers Continued

Counterexample

Suppose someone says “all comedians are funny,” and you disagree, how would you refute the statement?

       \(\forall x, C(x)\rightarrow F(x)\)

Refute by negating:

       \(\neg\forall x, C(x)\rightarrow F(x)\)
     \(\equiv\exists x, \neg(C(x)\rightarrow F(x))\)
     \(\equiv\exists x, \neg(F(x)\vee\neg C(x))\)
     \(\equiv\exists x, \neg F(x)\wedge C(x))\)

Your immediate response: _(some unfunny comedian)__

This evidence of refutation is called a _counterexample_.

Applies specifically to refuting \(\forall\)

Witness

Suppose someone says “there is a funny comedian,” and you disagree, how could they justify their statement?

       \(\exists x, C(x)\wedge F(x)\)

Their immediate response: _(a universally funny comedian – Robin Williams)_

This evidence of justification is called a _witness_.

Applies specifically to defending an \(\exists\)

Aside: Why not \(\exists x, C(x)\rightarrow F(x)\)?

Code Patterns

Nested Quantifiers

It will not surprise you that we can use multiple quantifiers:

 

     \(\forall x, \forall y \text{ Knows}(x,y)\)

In pairs, complete the second worksheet.

Words and Ideas

Quantifiers – “For all” and “There exists”

Counterexample and Witness

Negating quantifiers

Nested Quantifiers