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# 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 p:

$25×4=100$

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. $25×4=100$ , 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: $\mathrm{cot}\left(x\right)=\frac{1}{\mathrm{tan}\left(x\right)}$

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.

Sets

Subsets

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