Question

(a) Show whether V3 is rational or irrational.
(b) Explain inclusion-exclusion principle with example.
(c) Make logic circuit for the following Boolean expressions:
(x'y'z) + (x'y'z)'
( x'yz) (x'yz') (xy'z)​
please help guys for my assignment

Answer 1

for (a)

Answer:

root 3 is irrational number

for (b)

The principle of inclusion and exclusion (PIE) is a counting technique that computes the number of elements that satisfy at least one of several properties while guaranteeing that elements satisfying more than one property are not counted twice.

EXAMPLE:

As a simple example of the use of the principle of inclusion–exclusion, consider the question:[7]

How many integers in {1,...,100} are not divisible by 2, 3 or 5?

Let S = {1,...,100} and P1 the property that an integer is divisible by 2, P2 the property that an integer is divisible by 3 and P3 the property that an integer is divisible by 5. Letting Ai be the subset of S whose elements have property Pi we have by elementary counting: |A1| = 50, |A2| = 33, and |A3| = 20. There are 16 of these integers divisible by 6, 10 divisible by 10, and 6 divisible by 15. Finally, there are just 3 integers divisible by 30, so the number of integers not divisible by any of 2, 3 or 5 is given by:

100 − (50 + 33 + 20) + (16 + 10 + 6) − 3 = 26.

for( c)

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