# Discrete Math: Truth Tables

1) Construct the Truth Table for each of the following Boolean expressions:

Are they equivalent expressions? Are they tautologies? Contradictions?

2) Find a Boolean expression involving x y which produces the following table:

3) Consider a statement of the form "if A then (B and C)". Assume you wish to disprove it. Then you need to provide a counterexample, that is, you need to exhibit an instance for which

a) B and C are false

b) A is true and both B and C are false

c) B is false or C is false

d) A is true and B is false, or A is true and C is false

e) A is false, B is false, and C is false

Indicate which of the above is the correct answer.

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#### Solution Preview

Please see the attached files.

We use the equivalency:

(1.1)

And two De-Morgan rules:

(1.2)

The truth tables for the basic operations:

F F F F

F T T F

T F T F

T T T T

1.

The first expression can be converted using De-Morgan rule (b):

(1.3)

The truth table is:

F F T F

F T F F

T F T T

T T F F

2.

The expression is simplified:

(1.4)

This is a simple OR operation, so the truth table is:

F F F

F T T

T F T

T T T

3.

The expression can be simplifies using De-Morgan rule (a):

(1.5)

This is a simple NAND operator. It ...

#### Solution Summary

The expert examines truth tables in discrete mathematics. Counterexamples are provided.