# Engineering Mathematics – Well Formed Formulas (WFF)

**Well-Formed Formula(WFF) **is an expression consisting of variables(capital letters), parentheses, and connective symbols. An expression is basically a combination of operands & operators and here operands and operators are the connective symbols.

**Below are the possible Connective Symbols:**

- ¬ (Negation)
- ∧ (Conjunction)
- ∨ (Disjunction)
- ⇒ (Rightwards Arrow)
- ⇔ (Left-Right Arrow)

**Statement Formulas**

**1. **Statements that do not contain any connectives are called **Atomic** or **Simple** statements and these statements in themselves are **WFFs**.

*For example,*

P, Q, R, etc.

**2. **Statements that contain one or more primary statements are called **Molecular** or **Composite** statements.

*For example, *

If P and Q are two simple statements, then some of the Composite statements which follow WFF standards can be formed are:

-> ¬P

-> ¬Q

-> (P ∨ Q)

-> (P ∧ Q)

-> (¬P ∨ Q)

-> ((P ∨ Q) ∧ Q)

-> (P ⇒ Q)

-> (P ⇔ Q)

-> ¬(P ∨ Q)

-> ¬(¬P ∨ ¬Q)

**Rules of the Well-Formed Formulas**

- A Statement variable standing alone is a
**Well-Formed Formula(WFF)**.*For example*– Statements like P, ∼P, Q, ∼Q are themselves Well Formed Formulas. - If ‘P’ is a WFF then ∼P is a formula as well.
- If P & Q are WFFs, then (P∨Q), (P∧Q), (P⇒Q), (P⇔Q), etc. are also WFFs.

**Example Of Well Formed Formulas:**

WFF | Explanation |
---|---|

¬¬P | By Rule 1 each Statement by itself is a WFF, ¬P is a WFF, and let ¬P = Q. So ¬Q will also be a WFF. |

((P⇒Q)⇒Q) | By Rule 3 joining ‘(P⇒Q)’ and ‘Q’ with connective symbol ‘⇒’. |

(¬Q ∧ P) | By Rule 3 joining ‘¬Q’ and ‘P’ with connective symbol ‘∧’. |

((¬P∨Q) ∧ ¬¬Q) | By Rule 3 joining ‘(¬P∨Q)’ and ‘¬¬Q’ with connective symbol ‘∧’. |

¬((¬P∨Q) ∧ ¬¬Q) | By Rule 3 joining ‘(¬P∨Q)’ and ‘¬¬Q’ with connective symbol ‘∧’ and then using Rule 2. |

**Below are the Examples which may seem like a WFF but they are not considered as Well-Formed Formulas:**

**(P)**, ‘P’ itself alone is considered as a WFF by Rule 1 but placing that inside parenthesis is not considered as a WFF by any rule.**¬P ∧ Q**, this can be either (¬P∧Q) or ¬(P∧Q) so we have ambiguity in this statement and hence it will not be considered as a WFF. Parentheses are mandatory to be included in Composite Statements.**((P ⇒ Q))**, We can say (P⇒Q) is a WFF and let (P⇒Q) = A, now considering the outer parentheses, we will be left with (A), which is not a valid WFF. Parentheses play a really important role in these types of questions.**(P ⇒⇒ Q)**, connective symbol right after a connective symbol is not considered to be valid for a WFF.**((P ∧ Q) ∧)Q)**, conjunction operator after (P∧Q) is not valid.**((P ∧ Q) ∧ PQ)**, invalid placement of variables(PQ).**(P ∨ Q) ⇒ (∧ Q)**, with the Conjunction component, only one variable ‘Q’ is present. In order to form an operation inside a parentheses minimum of 2 variables are required.