Logic

De Morgan's Law, types of statements, notations

Definition: Statement/proposition

A sentence that’s either true or false but never both

Definition: Universal statement

States something is true for all elements in a set. Does not have to be true or false

ex/ All cats are black

Definition: Existential statement

At least one thing is true

ex/ Some cats are black

Definition: Compound statement

2 or more statements combined

ex/ Some cats are black, some cats are brown

Definition: Conditional Statement

Depending on given information, statement changes

ex/ If the animal is a cat, it is black

What is a conjunction?

An And statement. P and Q are both true

• Both conditions must be true to yield a true for the overall term

• Notation: P ^ Q

What is a disjunction?

An Or/Either statement

• Only one of the statements must be true to yield true

• Notation: P v Q

What is a negation?

Given a condition, it is NOT the condition

• Only yields true when the original condition is false

• Notation: if given P, then -P is negation

What is logical equivalence?

If both statements in a compound statement have the same truth table, they have the same outcome

What is De Morgan’s law?

• If you negate an AND statement, it becomes and OR statement

• If you negate an OR statement, it becomes and AND statement

• If they have the same truth table → logical equivalence

Definition: Tautology

A statement that is always true

ex/ It will either rain or it won’t rain

• Notation: p V -p

• This can be a type of truth table

Definition: Contradiction

A statement that is always false

ex/ It will rain and not rain

• Notation: p ^ -p

• This can be used to describe a truth table