Engineering Mathematics Well Formed Formulas (WFF) (original) (raw)

Last Updated : 21 Jun, 2024

**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.

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**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:

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:

  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.