Disjunction

In symbolic logic, a disjunction is defined as a compound statement formed by combining two simple statements with the word "or.' As you might expect, mathematicians generally don't write out "or" when they write a disjunction. Instead, they use a ∨ symbol between the statements they're combining. For example, A v B would link statement A and statement B with an 'or".

In this article, we'll explore how to work with disjunctions. Let's get started!

The properties of disjunction

We'll need some sample statements to practice with, so here are three:

Statement q: A trapezoid has two pairs of opposite sides parallel.

Statement r: The length of a circle's diameter is twice its radius.

It's worth checking if these are true before we start introducing disjunctions. $25\times 4=100$ , so statement p is true. Trapezoids do not have two pairs of opposite sides parallel, so statement q is false. A circle's radius measures half of its diameter, so statement r is also false.

A disjunction is considered true if any of the included statements are true.

$p\vee q$ is a true disjunction because statement p is true, even though statement q is false.

$q\vee r$ is a true disjunction because statement r is true, even though statement q is false.

The following truth table illustrates the truth value of $p\vee q$ based on the truth value of p and q:

Logic Table

$\begin{array}{ccc}p& q& p\vee q\\ T& T& T\\ T& F& T\\ F& T& T\\ F& F& F\end{array}$

When in doubt, remember that "or" makes it easier for a compound statement to be truthful than "and."

Practice questions on disjunction

a. If statement A reads that plant cells have cell walls and statement B reads that animal cells have mitochondria, what is the $A\vee B$ truth value of?

Statements A and B are both true: plant cells have cell walls and animal cells have mitochondria. Therefore, combining them with a ∨ symbol results in a true disjunction.

b. Consider the following statements:

w: Water boils at 200 degrees Fahrenheit

j: James K. Polk was President of the United States

m: John Milton wrote Romeo and Juliet

t: $\cot \left(x\right)=\frac{1}{\tan \left(x\right)}$

Write an example of both a true and false disjunction using the statements above.

The first step is verifying the truth value of each individual statement. Water boils at 212 degrees, so statement w is false. James K. Polk was President of the United States, so statement j is true. William Shakespeare wrote Romeo and Juliet, not Milton, so statement m is false. If you remember your trigonometric ratios, you know that statement t is true.

Next, we have to remember that a disjunction is true if either statement is true. We only have 2 false statements above, so $\left(w\vee m\right)$ must be our example of a false disjunction. For our true disjunction, we can use both of the true statements $\left(j\vee t\right)$ or either one of them in combination with a false statement $\left(j\vee t,m\vee t,\text{etc.}\right)$ . We now have an example of each!

c. Consider the following statements:

p: A century lasts 100 years

q: An hour contains 60 seconds

What is the truth value of $p\vee q$ ?

First, let's evaluate the truth value of each statement individually. A century indeed lasts 100 years, so statement p is true. However, an hour does not contain 60 seconds, so statement q is false. $p\vee q$ is true if either p or q is true, and statement p is. Therefore, $p\vee q$ is true.

d. Consider the following statements:

a: Mikey is 10 years old and his older brother is 8

b: New York City is located in the state of Nebraska

What is the truth value of $a\vee b$

A disjunction is true if either of the included statements is true, so let's look at our statements. Mikey can't be older than his older brother, so statement a is false. New York City is not located in the state of Nebraska, so statement b is also false. Since both statements are false, $a\vee b$ is also false.

Topics related to the Disjunction

Sets

Subsets

Operations on Sets

Flashcards covering the Disjunction

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Disjunction

Introduction to Proofs Practice Tests

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