## Boolean Function

Question 1 |

independent of one variable | |

independent of two variables | |

independent of three variable | |

dependent on all the variables |

Question 1 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 2 |

1 | |

a’ + b’ + c’ + d’ | |

a’ + b + c’ + d’ | |

a’ + b’ + c + d’ |

Question 2 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 3 |

XOR, AND | |

XOR, XOR | |

OR, OR | |

OR, AND |

Question 3 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 4 |

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 4 Explanation:

(A) XOR and NOT gates can only make XOR and XNOR which are not functionally complete.

(B) 2×1 multiplexer is functinally complete provided we have external 1 and 0 available.

(C) XOR can be used to make a NOT gate (a⊕1=a') and (AND, NOT) is functionally complete. Again this required external 1.

(D) We cannot derive NOT gate here. So not functionally complete.

Hence, options (B) and (C) are true provided external 1 and 0 are available.

(B) 2×1 multiplexer is functinally complete provided we have external 1 and 0 available.

(C) XOR can be used to make a NOT gate (a⊕1=a') and (AND, NOT) is functionally complete. Again this required external 1.

(D) We cannot derive NOT gate here. So not functionally complete.

Hence, options (B) and (C) are true provided external 1 and 0 are available.

Question 5 |

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

4 | |

17 | |

256 | |

65,536 |

Question 5 Explanation:

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

There are 5 questions to complete.