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)
1. Statements that do not contain any connectives are called Atomic or Simple statements and these statements in themselves are WFFs.
P, Q, R, etc.
2. Statements that contain one or more primary statements are called Molecular or Composite statements.
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) ∧ 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:
|¬¬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.