When working with logic in discrete math appliations there are a plethora of rules you can use for working with the *well formed formulas*. Remembering them all can be a daunting task, which is why I like to have a cheat sheet available. As such, here’s a simple one that I like to use when working with these problems.

# Equivalence Rules

Expression | Equivalent To | Name of the Rule |
---|---|---|

Commutative - comm. | ||

Commutative - comm. | ||

Associative - ass. | ||

Associative - ass. | ||

DeMorgan’s Law | ||

DeMorgan’s Law | ||

Implication - imp. | ||

Double Negation - dn. | ||

Definition of Equivalence - equ. |

# Inference Rules

Expression | Can Derive | Name of the Rule |
---|---|---|

Modus Ponens - mp. | ||

Modus Tollens - mt. | ||

Conjunction - con. | ||

Simplification - simp. | ||

Addition - add. | ||

Hypothetical Syllogism - hs | ||

Disjunctive Syllogism - ds. | ||

Contraposition - cont. | ||

Contraposition - cont. | ||

Self Reference - self. | ||

Self Reference - self. | ||

Exportation - exp. | ||

Inconsistency - inc. |

# Derivation Rules

The general workflow for using derivation rules is:

- Strip off the quantifiers
- Work with the independent well formed formulas
- Insert the quantifiers back in

**Universal Instantiation - ui.**: From we can deduce .

*Note: t must not already appear as a variable in the expression for .*

**Existential Instantiation - ei.**: From we can deduce .

*Note: t must be introduced for the first time. As such, you will want to do these early in proofs.*

**Universal Generalization - ug.**: From we can deduce .

*Notes:*

*hasn’t been deduced by existential instantiation from any hypothesis in which x was a free variable.**hasn’t been deduced by existential instantiation from another wff in which x was a free variable.*

**Existential Generalization - eg.**: From we can deduce .

*Note: x must not appear in .*