# Engineering Mathematics – Well Formed Formulas (WFF)

• Difficulty Level : Easy
• Last Updated : 17 Dec, 2021

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:

1. ¬ (Negation)
2. ∧ (Conjunction)
3. ∨ (Disjunction)
4. ⇒ (Rightwards Arrow)
5. ⇔ (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

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

### Example Of Well Formed Formulas:

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

1. (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.
2. ¬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.
3. ((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.
4. (P ⇒⇒ Q), connective symbol right after a connective symbol is not considered to be valid for a WFF.
5. ((P ∧ Q) ∧)Q), conjunction operator after (P∧Q) is not valid.
6. ((P ∧ Q) ∧ PQ), invalid placement of variables(PQ).
7. (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.
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