# Propositions and Logical Operations

## 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
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.

1. ¬ (not)
2. ∧ (and)
3. ∨ (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 examples
Example #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)

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