Source: https://xkcd.com/1856/

## All posts in category **Discrete mathematics**

# Existence Proofs

*Posted by John Quintanilla on May 29, 2020*

https://meangreenmath.com/2020/05/29/existence-proofs/

# How To Use Facebook Emoji to Respond to a Mathematical Proof

*Posted by John Quintanilla on May 22, 2020*

https://meangreenmath.com/2020/05/22/how-to-use-facebook-emoji-to-respond-to-a-mathematical-proof/

# Adding by a Form of 0 (Part 3)

As part of my discrete mathematics class, I introduce my freshmen/sophomore students to various proof techniques, including proofs about sets. Here is one of the examples that I use that involves adding and subtracting a number *twice* in the same proof.

**Theorem**. Let be the set of even integers, and define

Then .

**Proof (with annotations)**. Before starting the proof, I should say that I expect my students to use the formal definitions of even and odd:

- An integer is even if for some integer .
- An integer is odd if for some integer .

To prove that , we must show that and . The first of these tends to trickiest for students.

*Part 1. *Let . By definition of even, that means that there is an integer so that .

To show that , we must show that for some odd integer . To this end, notice that . Thus, we must show that is an odd integer, or that can be written in the form . To do this, we add and subtract 1 a second time:

.

By the closure axioms, is an integer. Therefore, is an odd number by definition of odd, and hence $n \in B$.

The above part of the proof can be a bit much to swallow for students first learning about proofs. For completeness, let me also include Part 2 (which, in my experience, most students can produce without difficulty).

*Part 2*. Let , so that for some odd integer . By definition of odd, there is an integer so that $m = 2k+1$. Therefore, . By the closure axioms, is an integer. Therefore, is even by definition of even, and so we conclude that .

For what it’s worth, this is the review problems for which I recorded myself talking through the solution for the benefit of my students.

In my opinion, the biggest conceptual barriers in this proof are these steps from Part 1:

.

These steps are undeniably awkward. Back in high school algebra, students would get points taken off for making the expression more complicated instead of simplifying the answer. But this is the kind of jump that I need to train my students to do so that they can master this technique and be successful in their future math classes.

*Posted by John Quintanilla on January 27, 2020*

https://meangreenmath.com/2020/01/27/adding-by-a-form-of-0-part-3/

# Predicate Logic and Popular Culture (Part 206): Jack Johnson

Let be the set of all things, let be the set of all times, let be the proposition “ is good,” and let be the proposition “ remains at time .” Translate the logical statement

.

This matches a line from “Mudfootball” by Jack Johnson.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

*Posted by John Quintanilla on January 6, 2020*

https://meangreenmath.com/2020/01/06/predicate-logic-and-popular-culture-part-206-jack-johnson/

# Predicate Logic and Popular Culture (Part 205): Bob Marley

Let be the set of all things, let be the proposition “ is a little thing,” and let be the proposition “ is going to be all right.” Translate the logical statement

.

This matches a line from “Three Little Birds” by Bob Marley.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

*Posted by John Quintanilla on January 3, 2020*

https://meangreenmath.com/2020/01/03/predicate-logic-and-popular-culture-part-205-bob-marley/

# Predicate Logic and Popular Culture (Part 204): Billy Joel

Let be the set of all times, and let be the proposition “She is a woman to me at time .” Translate the logical statement

.

This matches a line from “She’s Always a Woman” by Billy Joel.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

*Posted by John Quintanilla on December 30, 2019*

https://meangreenmath.com/2019/12/30/predicate-logic-and-popular-culture-part-204-billy-joel/

# Predicate Logic and Popular Culture (Part 203): Bill Withers

Let be the set of all people, let be the set of all times, let be the proposition “ has pain at time ,” and let be the proposition “ has sorrow at time .” Translate the logical statement

.

This matches a line from “Lean on Me.” Note: while I think the translation above matches the intent of the song, a case could be made that, literally rendered, the “there exists” symbols should come first — that there’s a single time that everyone has pain at that one time.

*Posted by John Quintanilla on December 27, 2019*

https://meangreenmath.com/2019/12/27/predicate-logic-and-popular-culture-part-203-bill-withers/

# Predicate Logic and Popular Culture (Part 202): The LEGO Movie

Let be the set of all things, let be the proposition “You’re part of a team,” let be the proposition “ is awesome,” and let be the proposition “ is cool.” Translate the logical statement

.

This matches the opening line of “Everything is Awesome!!!” from The LEGO Movie.

*Posted by John Quintanilla on December 20, 2019*

https://meangreenmath.com/2019/12/20/predicate-logic-and-popular-culture-part-202-the-lego-movie/

# Predicate Logic and Popular Culture (Part 201): Hamilton

Let be the set of all times, let time 0 be now, and let be the proposition “I like the quiet at time .” Translate the logical statement

.

This matches a line from “It’s Quiet Uptown” from the hit musical *Hamilton*.

*Posted by John Quintanilla on December 16, 2019*

https://meangreenmath.com/2019/12/16/predicate-logic-and-popular-culture-part-201-hamilton/

# Predicate Logic and Popular Culture (Part 200): Spider-Man

Let be the set of all times, and let be the proposition “I will rest at time ,” and let be the proposition “You are unmasked and eliminated at time .” Translate the logical statement

.

This matches a line by J. Jonas Jameson in the 1990s Spider-Man cartoons.

*Posted by John Quintanilla on December 13, 2019*

https://meangreenmath.com/2019/12/13/predicate-logic-and-popular-culture-part-200-spider-man/