- Conjunctions and disjunctions are useful tools for building algorithms.
- They enable you to combine propositions.
- Truth tables are a fast way to find solutions.
- Analogies can help you to remember the results.

Dive into machine learning, and you'll come across algorithms that include conjunctions** and disjunctions**. For example, you might come across a set of conjunctive rules in a hypothesis space (the set of all functions a model can return) or create a learning algorithm that builds a conjunction using similar features.

Conjunctions and Disjunctions are one way to **combine propositions into more complex ones.** Propositions [noterm] are statements that are either true or false. For example, "2 is greater than 3" or "10 + 10 = 21." Some statements like "He is a great swimmer" or "How are you today" don't have true or false answers and so aren't propositions. Once you have a set of propositions, you can combine them in various ways, including:

**Conjunction (and, &, ∧)**: combine (add) propositions.**Disjunction****(or):**choose (select) propositions.

Many outcomes for these two simple statements are possible. You can quickly find solutions using truth tables.

In order to account all the possible combinations of truth values for two statements p and q, we can create a four-row truth table:

A key fact from the table: a conjunction is true only if all the variables in it are true. If you're familiar with the simple model theory of eye color (where Brown B is dominant over blue b) [1], one way to **make sense of the table results** is just to remember a single fact: "False" is dominant over "true". In order for *true* to appear, it must be paired (t, t). But if *False* shows in any position, it dominates truth; It will result in an F even when paired with true (F, t) or (t, F).

In that case, an **empty conjunction is always defined as true**. [2] The eye color analogy obviously doesn't work here, but an old saying *does* work:

*One bad apple spoils the bunch.*

Imagine you have a basket that you're going to fill with varieties of good (true) and bad (False) apples. If you have a single false statement, everything in the basket is tainted (i.e. False). But if your basket is empty (a.k.a. the empty truth table), then it hasn't been filled with apples yet . You have no reason to assume that your basket is going to be filled with good apples.* *Unless you're very pessimistic, in which case perhaps data science isn't the career for you!

Similarly, a truth table can find solutions for disjunctions. The format is the same but the results are slightly different:

Here's my analogy for remembering the results here. It's similar to the "bunch of apples" analogy except here we are given a choice: apple P **or** apple Q. So, given that basket of apples where some are rotten, which would you choose? Every time, you would choose the good (a.k.a. true) apple. In the last row, you have two bad apples so you have no choice but to pick one of those.

The **empty disjunction is defined as false.** Back to our basket analogy, the "OR" here is you being forced to choose between good apples OR bad apples that are already in the basket. The basket is empty when everyone has chosen their apples. You're left with a basket that has, unfortunately, mold in it from those bad apples. So you're left with bad (False) residue.

Let's take two statements:

**P**: There are 99 cents in $1

**Q**: The dollar ($) is US currency.

We want to know: *what is the conjunction of P and Q?*

**Solution**:

**Step 1.** Construct a truth table. This is an "and" question, so create a conjunction truth table.

**Step 2:** Determine whether the statements are true or false. For this example, P is False and Q is true.

**Step 3:** Refer to the line that reflects whether the statements are true or False. The third line (F, t = F) is the correct solution.

**References**

Table pictures by author.

© 2021 TechTarget, Inc. Powered by

Badges | Report an Issue | Privacy Policy | Terms of Service

**Most Popular Content on DSC**

To not miss this type of content in the future, subscribe to our newsletter.

- Book: Applied Stochastic Processes
- Long-range Correlations in Time Series: Modeling, Testing, Case Study
- How to Automatically Determine the Number of Clusters in your Data
- New Machine Learning Cheat Sheet | Old one
- Confidence Intervals Without Pain - With Resampling
- Advanced Machine Learning with Basic Excel
- New Perspectives on Statistical Distributions and Deep Learning
- Fascinating New Results in the Theory of Randomness
- Fast Combinatorial Feature Selection

**Other popular resources**

- Comprehensive Repository of Data Science and ML Resources
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- 100 Data Science Interview Questions and Answers
- Cheat Sheets | Curated Articles | Search | Jobs | Courses
- Post a Blog | Forum Questions | Books | Salaries | News

**Archives:** 2008-2014 |
2015-2016 |
2017-2019 |
Book 1 |
Book 2 |
More

**Most popular articles**

- Free Book and Resources for DSC Members
- New Perspectives on Statistical Distributions and Deep Learning
- Time series, Growth Modeling and Data Science Wizardy
- Statistical Concepts Explained in Simple English
- Machine Learning Concepts Explained in One Picture
- Comprehensive Repository of Data Science and ML Resources
- Advanced Machine Learning with Basic Excel
- Difference between ML, Data Science, AI, Deep Learning, and Statistics
- Selected Business Analytics, Data Science and ML articles
- How to Automatically Determine the Number of Clusters in your Data
- Fascinating New Results in the Theory of Randomness
- Hire a Data Scientist | Search DSC | Find a Job
- Post a Blog | Forum Questions

## You need to be a member of Data Science Central to add comments!

Join Data Science Central