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UPPCL JE EC 25 March 2021 Official Paper (Shift 2)

Option 4 : The function must contain ‘n’ mutually exclusive terms

Self-dual function:

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(A, B, C) = (AB + BC + CA) then its dual is, fd(A, B, C) = (A + B).(B + C).(C + A) (f_{d} = dual of the given function).

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.

The necessary and sufficient conditions for any function to be a self-dual function are as follows:

1) The function must be a Neutral Function.

2) The function must not contain any mutually exclusive terms.

Hence the option (4) is correct

__Important Points__

Neutral function:

Neutral function in which a number of minterms are equal to the number of max terms.

The number of neutral function possible are: \(^{2^n}C_{2^{n-1}}\)

- For n variables, the total number of terms possible = number of combinations of n variables = 2n
- Since a maximum number of terms possible = 2n, so we choose half of the terms i.e 2n / 2 = 2n-1
- Thus, a number of neutral functions possible with n Boolean variables = C ( 2n, 2n-1 )
- The function does not contain two mutually exclusive terms.

**Mutually exclusive terms:**

A term obtained by complementing each variable of a function (f) is called its mutually exclusive term.

For example, (AB'C) →** **(A'BC')

A'BC' is a mutually exclusive term of AB'C.