The dual of a statement formula is obtained by replacing ∨ by ∧ , ∧ by ∨ , T by F F by T. A dual is obtained by replacing T (tautology) by (contradiction), F and, by T.

**Mathematical Logic**

**Duality**

The **dual** of a statement
formula is obtained by replacing ∨ by ∧ , ∧ by ∨ , *T* by *F* *F
*by* T***.*** *A dual is obtained by
replacing **T*** ***(tautology)*** *by* ***(contradiction)**, **F** and,* *by*
***T.**

(1) The
symbol ¬
is not changed while finding the dual.

(2) Dual
of a dual is the statement itself.

(3) The
special statements **T** (tautology) and
**F** (contradiction) are duals of each
other.

(4)* T *is changed to* F *and vice-versa.

** **

If a
compound statement *S*_{1}
contains only ¬, ∧
, and ∨ and statement *S*_{2} arises from *S*_{1}
by replacing ∧ by ∨ , and, ∨ by ∧ then *S*_{1} is a tautology if and only if *S*_{2} is a contradiction.

(i) The
dual of ( *p* ∨ *q*)
∧ (*r* ∧ *s* ) ∨ **F **is ( *p* ∧ *q*)
∨ ( *r* ∨ *s* ) ∧ **T**.

(ii) The
dual of *p* ∧ [¬ *q*
∨ ( *p* ∧ *q*) ∨ ¬ *r*
] is *p* ∨ [¬ *q*
∧ ( *p* ∨ *q*) ∧ ¬ *r*
].

Tags : Discrete Mathematics | Mathematics , 12th Maths : UNIT 12 : Discrete Mathematics

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12th Maths : UNIT 12 : Discrete Mathematics : Mathematical Logic: Duality | Discrete Mathematics | Mathematics

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