# Self Dual functions in Digital Logic

A function is said to be **Self dual** if and only if its dual is equivalent to the given function, i.e., if a given function is **f(X, Y, Z) = (XY + YZ + ZX)** then its dual is, **fd(X, Y, Z) = (X + Y).(Y + Z).(Z + X)** (fd = dual of the given function) = **(XY + YZ + ZX)**, it is equivalent to the given function. So function is self dual.

In a dual function:

- AND operator of a given function is changed to OR operator and vice-versa.
- A constant 1 (or true) of a given function is changed to a constant 0 (or false) and vice-versa.

A Switching function or Boolean function is said to be Self dual if :

- The given function is neutral i.e., (number of min terms is equal to the number of max terms).For more about min term and max term (see Canonical and standard Form).
- The function does not contain two mutually exclusive terms.

**Note:** Mutually exclusive term of XYZ is (X’Y’Z’) i.e, compliment of XYZ. So, two mutually exclusive terms are compliment of each other.

**Example:**

SL NO. | X | Y | Z |
---|---|---|---|

0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 |

2 | 0 | 1 | 0 |

3 | 0 | 1 | 1 |

4 | 1 | 0 | 0 |

5 | 1 | 0 | 1 |

6 | 1 | 1 | 0 |

7 | 1 | 1 | 1 |

In the above table, Mutually exclusive terms are:

(0,7), (1,6), (2,5), (3,4)

**Explanation:**

- Compliment of (000) i.e, 0 is (111) i.e, 7 so, (0, 7 are mutually exclusive to each other.)
- Compliment of (001) i.e, 1 is (110) i.e, 6 so, (1, 6 are mutually exclusive to each other.)
- Compliment of (010) i.e, 2 is (101) i.e, 5 so, (2, 5 are mutually exclusive to each other.)
- Compliment of (011) i.e, 3 is (100) i.e, 4 so, (3, 4 are mutually exclusive to each other.)

Now, lets check number of Self dual functions possible for a given function.

Let, a function has **n** variables then,

Number of pairs possible = 2^{n}/2 = 2^{(n-1)}

Therefore, number of Self dual functions possible with **n** variables

= 2^{2(n-1)}

There are 2 possibilities for each pair.

**Example:** What is total number of self dual of a function which has 3 variables X, Y and Z ?

= 2^{2(3-1)}= 2^{22}= 2^{4}= 16

**Note:**

- Every Self dual function is neutral but every neutral function is not Self dual.
- Self duality is closed under compliment i.e, compliment of a Self dual function is also Self dual.

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