## Well-Formed Formulas

** Well-Formed Formulae(WFF)**

In discrete mathematics, well-formed formula is defined as a statement formula is an expression which is a string consisting of variables (capital letters with subscripts or without subscripts), parenthesis, and connective symbols. Not every string of these symbols is a formula, We shall now give a recursive definition of a statement formula, often called a well-formed formula (WFF). A well-formed formula can be generated by the following rules

- A statement variable standing alone is a well-formed formula.
- If A is a Well-Formed Formula, then ~A is a well-formed formula.
- If A and B are well-formed formulas, then (A∧B), (A∨B), (A
**→**B), and (A**⇔**B) are well formed formulas. - A string of symbols containing the statement variables, connective and parenthesis is a well formed formulas if and only if it can be obtained by finitely many applications of the rules 1, 2 and 3.

The following are well formed formulas:

~(P∧Q), ~(P∨Q), (P**→** (Q∨R)), (P**→**(Q**→**R))

The following are not well-formed formulas:

1.~P**→**Q, In this obviously P and Q are well formed formulas A well formed formula would be either (~P**→**Q) or ~(P**→**Q).

2.(P**→**Q)** →** (∧Q), this is not a well formed formula because ∧Q is not.

3.(P∧Q. here note that (P∧Q) is a well formed formula.

In this we shall not discuss these conventions here. It is possible to introduce some conventions so that the number of parentheses used can be reduced.

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