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Discrete Math Problems: Boolean Algebra

1. Let x, y be elements in the Boolean algebra B. Prove that x = y if and only if xy + xy = 0.

2. a. How many rows are needed to construct the (function) table for a Boolean function of n variables?
b. How many different Boolean functions of n variables are there?

3. Let g: B4 →B be defined by g(w, x, y, z) = (wz + xyz)(x + x yz).
a. Find the d.n.f. and c.n.f. for g.
b. Write g as a sum of minterms an d as a product of maxterms (utilizing binary labels).

4. Obtain a minimal-product-of-sums representation for f (w, x, y, z) IIM(0, 1, 2, 4, 5, 10, 12, 13, 14).

5. Let f, g: B5 →B be Boolean functions, where f =∑m(1, 2, 4, 7, x) and g = ∑m(0, 1, 2, 3, y, z, 16, 25). If f ≤ g, what are x, y, z?

Solution Preview

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1. Let x, y be elements in the Boolean algebra B. Prove that x = y if and only if xy + xy = 0.
If x=y, then xy+ xy = x2+x2= x+x=0
If xy + xy = 0, then xy = 0 and then x=y = 0.

2. a. How many rows are needed to construct the (function) table for a Boolean function of n variables?
There are Boolean functions.

b. How many different Boolean functions of n variables are there?
There are different Boolean functions.

3. Let g: B4 →B be defined by g(w, x, y, z) = (wz ...

Solution Summary

Discrete mathematics for Boolean algebra are discussed. How many rows are needed to construct the function tables for a Boolean function of n variables is provided.

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