How do you prove the equivalence or contradiction of two control logic statements using truth tables?

1 What are control logic statements?

Control logic statements are expressions that use logical operators and variables to represent the input and output conditions of a system or a device. For example, a control logic statement for a traffic light could be: R = A AND B, where R is the output for the red light, A is the input for the timer, and B is the input for the sensor. Logical operators include AND, OR, NOT, XOR, and others, and they follow certain rules and properties.

2 What are truth tables?

Truth tables are tables that show all the possible combinations of input and output values for a control logic statement or a circuit. They are useful tools to visualize and test the functionality and behavior of a system or a device. For example, a truth table for the traffic light control logic statement could look like this:

| A | B | R |

|---|---|---|

| 0 | 0 | 0 |

| 0 | 1 | 0 |

| 1 | 0 | 0 |

| 1 | 1 | 1 |

Where 0 represents false and 1 represents true.

3 What are boolean algebra theorems?

Boolean algebra theorems are rules and principles that control the manipulation and simplification of control logic statements and circuits. These theorems are based on the axioms and properties of boolean algebra, such as identity, commutativity, associativity, distributivity, complementarity, and others. De Morgan's theorem states that NOT (A AND B) = NOT A OR NOT B, and NOT (A OR B) = NOT A AND NOT B. The double negation theorem is NOT (NOT A) = A. The idempotent theorem is A AND A = A, and A OR A = A. The involution theorem is NOT (NOT A) = A. Finally, the absorption theorem states that A AND (A OR B) = A, and A OR (A AND B) = A.

4 How to use truth tables to prove theorems?

One way to prove the validity of a boolean algebra theorem is to use truth tables to compare the input and output values of the two sides of the equation. If the two sides have the same output values for all possible input values, then the theorem is true. For example, to prove De Morgan's theorem for NOT (A AND B) = NOT A OR NOT B, we can construct a truth table like this:

| A | B | NOT (A AND B) | NOT A | NOT B | NOT A OR NOT B |

|---|---|----------------|-------|-------|----------------|

| 0 | 0 | 1 | 1 | 1 | 1 |

| 0 | 1 | 1 | 1 | 0 | 1 |

| 1 | 0 | 1 | 0 | 1 | 1 |

| 1 | 1 | 0 | 0 | 0 | 0 |

Since the output values of NOT (A AND B) and NOT A OR NOT B are the same for all input values, we can conclude that the theorem is true.

5 How to use truth tables to prove equivalence or contradiction?

Similarly, we can use truth tables to prove the equivalence or contradiction of two control logic statements. If the two statements have the same output values for all possible input values, then they are equivalent. If they have opposite output values for all possible input values, then they are contradictory. For example, to prove the equivalence of A XOR B and (A AND NOT B) OR (NOT A AND B), we can construct a truth table like this:

| A | B | A XOR B | A AND NOT B | NOT A AND B | (A AND NOT B) OR (NOT A AND B) |

|---|---|---------|-------------|-------------|---------------------------------|

| 0 | 0 | 0 | 0 | 0 | 0 |

| 0 | 1 | 1 | 0 | 1 | 1 |

| 1 | 0 | 1 | 1 | 0 | 1 |

| 1 | 1 | 0 | 0 | 0 | 0 |

Since the output values of A XOR B and (A AND NOT B) OR (NOT A AND B) are the same for all input values, we can conclude that they are equivalent.