In mathematical logic, words have precise meanings. Logic attempts to show truthful conclusions emerging from truthful premises, or it identifies falsehoods reliably. Mathematical and logical statements are joined with connectors; conjunctions and disjunctions are two types of logical connectors.

## Logic Statements

With logic, statements can be labeled as true or false, such as:

1. All numbers are integers
2. Some negative numbers are integers
3. Squares are rectangles
6. Rectangles have four sides

Clearly, some of those six statements are false, but the point is, they are testable claims. They do not express opinion. Contrast them with, say, "I like cheeseburgers," which shows an opinion.

## Logic Connectors

Statements are often symbolized with the letters p and q. They are strung together using connectors, so you can combine ideas using "and," or "or" between statements. Two statements joined with connectors create a compound statement. Joining logical statements is not the same as stringing together ideas in ordinary English conversation. Compare:

• I like cheeseburgers and my friend enjoys banana milkshakes
• All numbers are integers and squares are rectangles

The first connected statements, a single compound statement, are opinions. The second compound statement is a logical statement (but the compound statement is false).

The two types of connectors are called conjunctions ("and") and disjunctions ("or"). Conjunctions use the mathematical symbol $\wedge$ and disjunctions use the mathematical symbol $\vee$.

## Conjunctions In Math

Joining two statements with "and" is a conjunction, which means both statements must be true for the whole compound statement to be true. Conjunctions are symbolized with the $\wedge$ character, so these two discrete statements can be combined in a compound statement:

• Statement p: Squares are rectangles
• Statement q: Rectangles have four sides
• Compound statement (in English): Squares are rectangles and rectangles have four sides.
• Compound statement (in mathematical symbols): p$\wedge$q

Only if both parts of the compound statement are true is the entire statement true.

## Disjunctions In Math

When the connector between two statements is "or," you have a disjunction. In this case, only one statement in the compound statement needs to be true for the entire compound statement to be true.

Let's look at our original statements again:

1. All numbers are integers
2. Some negative numbers are integers
3. Squares are rectangles
6. Rectangles have four sides

If we link one true and one false statement into a compound statement using the connector "or," (symbolized by $\vee$) we still have a true compound statement:

• p: Squares are rectangles
• q: Quadrilaterals have 11 sides
• p$\vee$q: All squares are rectangles or quadrilaterals have 11 sides

## Conjunction And Disjunction Examples

Here are four other compound statements taken from our original statements. Determine the symbols and if the compound statements are true or false:

• p: Some negative numbers are integers
• q: Squares are rectangles
• Some negative numbers are integers and squares are rectangles.

Did you say p$\wedge$q, and did you rate this as true? Both statements are true, so the compound statement joined by "and" is true.

• p: Some quadrilaterals are parallelograms
• q: Quadrilaterals have 11 sides

Did you say p$\vee$q, and rate this compound statement as true? Though quadrilaterals do not have 11 sides, the conjunction "or" makes the compound statement true since some quadrilaterals are parallelograms.

• p: Quadrilaterals have 11 sides
• q: Rectangles have four sides
• Quadrilaterals have 11 sides and rectangles have four sides.

Did you say p$\wedge$q? More important, did you say this compound statement was false? Since quadrilaterals do not have 11 sides, the conjunction is false.

• p: All numbers are integers
• q: Squares are rectangles
• All numbers are integers or squares are rectangles.

Did you write p$\vee$q? Did you say this is false, since both sides of the compound statement are false?

Conjunctions and disjunctions are ways of joining logical statements, with every joined, compound statement either true or false. For conjunctions, both statements must be true for the compound statement to be true. For disjunctions, only one statement needs to be true for the compound statement to be true.

### Next Lesson: Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
Tutors online

### Find a math tutor in Ashburn, VA

Learn faster with a math tutor. Find a tutor locally or online.