## Boolean-Functions

Question 1 |

A set of Boolean connectives is functionally complete if all Boolean functions can be synthesized using those. Which of the following sets of connectives is NOT functionally complete?

EX-NOR | |

implication, negation | |

OR, negation | |

NAND |

Question 1 Explanation:

→ EX-NOR is not functionally complete.

→ NOR and NAND are the functionally complete logic gates, OR, AND, NOT only logic gate can be implemented by using them.

→ And (Implication, Negation) is also functionally complete.

→ NOR and NAND are the functionally complete logic gates, OR, AND, NOT only logic gate can be implemented by using them.

→ And (Implication, Negation) is also functionally complete.

Question 2 |

independent of one variable | |

independent of two variables | |

independent of three variable | |

dependent on all the variables |

Question 2 Explanation:

f(A, B, C, D) = Σ(2, 3, 6, 7, 8, 9, 10, 11, 12, 13)

Independent of one variable '0'.

Independent of one variable '0'.

Question 3 |

1 | |

a’ + b’ + c’ + d’ | |

a’ + b + c’ + d’ | |

a’ + b’ + c + d’ |

Question 3 Explanation:

(a⋅b)⋅c + (a'⋅c)⋅d + (b⋅c)⋅d + a⋅d

= ((ab)'c)' + ((a'c)'d)' + ((bc)'d)' + (ad)'

= ab + c' + a'c + d' + bc + d' + a' + d'

= ab + c' + a'c + bc + a' + d'

= ab + c' + bc + a' + d'

= b + c' + bc + a' + d'

= a' + b + c' + d'

= ((ab)'c)' + ((a'c)'d)' + ((bc)'d)' + (ad)'

= ab + c' + a'c + d' + bc + d' + a' + d'

= ab + c' + a'c + bc + a' + d'

= ab + c' + bc + a' + d'

= b + c' + bc + a' + d'

= a' + b + c' + d'

Question 4 |

XOR, AND | |

XOR, XOR | |

OR, OR | |

OR, AND |

Question 4 Explanation:

Thus we have OR and AND which gives different outputs on (0,0) and (1,1).

The encodes can be hence select from the two and decide output of the function according to x.

Question 5 |

Which of the following sets of component(s) is/are sufficient to implement any arbitrary Boolean function?

XOR gates, NOT gates | |

2 to 1 multiplexors | |

AND gates, XOR gates | |

Three-input gates that output (A⋅B) + C for the inputs A⋅B and C | |

Both B and C |

Question 5 Explanation:

(A) Not complete because, XOR can be used to make only NOT gate and NOT gate is already available. Hence not complete.

(B) 2 to 1 multiplexors is functionally complete.

(C) XOR gate can be used to make a NOT gate. So, (AND, NOT) is functionally complete.

(D) With given gates and inputs NOT gate cannot be derived.

Hence, not complete.

(B) 2 to 1 multiplexors is functionally complete.

(C) XOR gate can be used to make a NOT gate. So, (AND, NOT) is functionally complete.

(D) With given gates and inputs NOT gate cannot be derived.

Hence, not complete.

Question 6 |

The total number of Boolean functions which can be realised with four variables is:

4 | |

17 | |

256 | |

65,536 |

Question 6 Explanation:

Total no. of Boolean functions which can be realized with four variables is:

There are 6 questions to complete.