## Propositions

In logic, a proposition is a statement that evaluates true or false, such as “all cars are blue” or “4 + 4 = 8.” A phrase such as “go outside” is a command and not a proposition because it does not have a true or false value. Propositional statements can be denoted by variables, such as `Q = "all cars are blue,"`

or `P = "4 + 4 = 8," `

which evaluate to `Q = False`

and `P = True`

.

## Compound Propositions

Compound propositions are the result of combining multiple propositions. For example, `K = Q and P`

. That compound proposition evaluates to false because, as we’ve seen, Q is false. We can make K evaluate to true by changing the compound proposition to `K = Q or P`

. The condition here is that only one variable must evaluate true for K to be true because we are saying “or” instead of “and.”

## Logic Operators

logical operators refer to operations that combine propositions, such as the “and” and “or” operators we’ve previously used. Below is a list of logical operators and their denotations in discrete mathematics.

**The conjunction operation** is denoted by ∧ and read as “and,” such as:

K = Q ∧ P

read as

K = Q and P

**The disjunction operation** reads as “or” and has two distinctions, the “exclusive or” and the “inclusive or.”

**The “inclusive or”** denoted by ∧ and reads as “or” dictates that a compound proposition is true if either statement is true.

**Inclusive-or example:**K = True

P = False

N = True

R = K ∧ P ∧ N

R evaluates to True when either K, P, or N is true.

**The “exclusive or”** denoted by ⊕ and reads as “or” dictates that only one operation must evaluate true for the compound proposition to be true.

**Exclusive-or example:**K = True

P = False

N = True

R = K ∧ P ∧ N

R evaluates to False when more than variables are true.

**The negation operation**, denoted by ¬ and read as “not,” reverses the value of a proposition. ¬Q is read as “not Q.”

Negation operation example:

P = True

T = False

Q = P ∧ T (Q evaluates to true because P is true)

Q = ¬P ∧ T (Q evaluates to false because both P and T are false)

The negation of ¬P reversed P’s value from True to False.

## Order of Operation

The order of operation for compound propositions is (¬) negation first, (∧) conjunction second, and (∨) disjunction last.

- ¬ (not)
- ∧ (and)
- ∨ (or)

Similar to the order of operation in mathematics, everything in parenthesis is applied first except that the negation operation (¬) always takes precedence.

**Order of operation examplesExample #1**

`F ∨ T ∧ T`

`F ∨ (T ∧ T)`

(Conjunction has precedence over disjunction)`F ∨ (T)`

(True and True evaluate to True)`T`

(False or True evaluate to True)**Example #2**

`¬F ∨ T ∧ T`

`(¬F) ∨ T ∧ T`

(Negation has precedence over conjunction and disjunction`(¬F) ∨ (T ∧ T)`

(Conjunction has precedence over disjunction)`(¬F) ∨ T`

(True and True evaluate to True)`T ∨ T `

(Negation of False evaluates to True)`T `

(True and True evaluate to True)