__NOT Rule__**In algebra, the negative of a negative is a positive and taking the inverse of an inverse returns the original value. Although the NOT gate does not have an equivalent in mathematical algebra, it operates in a**

**similar manner. If the boolean inverse of a boolean inverse is taken, the original value results.**

**This is proven with a truth table.**

**Since the first column and the third column have the same pattern of ones and zeros, they must be equivalent. Figure shows this rule in schematic form.**

__OR Rules__**If an input to a logic gate is a constant 0 or 1 or if the same signal is connected to more than one input of a gate, a simplification of the expression is almost always possible. This is true for the OR gate as is shown with the following four rules for simplifying the OR function.**

**First, what happens when one of the inputs to an OR gate is a constant logic 0? It turns out that the logic 0 input drops out leaving the remaining inputs to stand on their own. Notice that the two columns**

**in the truth table below are equivalent thus proving this rule.**

**Rule: A + 0 = A**

**A A + 0**

**0 0+0 = 0**

**1 1+0 = 1**

**Rule: A + 1 = 1**

**A A + 1**

**0 0+1 = 1**

**Rule: A + A = A**

**A A+ A**

**0 0+0 = 0**

**1 1+1 = 1**

**Rule: A + A^ = 1**

**A A + A^**

**0 0+1 = 1**

**1 1+0 = 1**

__AND Rules__**Just as with the OR gate, if either of the inputs to an AND gate is a constant (logic 0 or logic 1) or if the two inputs are the same or inverses of each other, a simplification can be performed. Let's begin with the case where one of the inputs to the AND gate is a logic 0. Remember that an AND gate must have all ones at its inputs to output a one. In this case, one of the inputs will always be zero forcing this AND to**

**always output zero. The truth table below shows this.**

**Rule: A · 0 = 0**

**A A · 0**

**0 0 · 0 = 0**

**1 1 · 0 = 0**

**Rule: A · 1 = A**

**A A · 1**

**0 0 · 1 = 0**

**1 1 · 1 = 1**

**Rule: A · A^= 0**

**A A · A^**

**0 0 · 1 = 0**

**1 1 · 0 = 0**

**Rule: A · A = A**

**A A · A**

**0 0 · 0 = 0**

**1 1 · 1 = 1**

__XOR Rules__**Now let's see what happens when we apply these same input conditions to a two-input XOR gate. Remember that a two-input XOR gate outputs a 1 if its inputs are different and a zero if its inputs are the**

**same. If one of the inputs to a two-input XOR gate is connected to a logic 0, then the gate's output follows the value at the second input. In other words, if the second input is a zero, the inputs are the same forcing the**

**output to be zero and if the second input is a one, the inputs are different and the output equals one.**

**Rule: A ⊕ 0 = A**

**A A ⊕ 0**

**0 0 ⊕ 0 = 0**

**1 1 ⊕ 0 = 1**

**Rule: A ⊕ A = 1**

**A A ⊕ A**

**0 0 ⊕ 1 = 1**

**1 1 ⊕ 0 = 1**

**Rule: A ⊕ A^ = 0**

**A A ⊕ A^**

**0 0 ⊕ 0 = 0**

**1 1 ⊕ 1 = 0**

**Rule: A ⊕ 1 = A**

**A A ⊕ 1**

**0 0 ⊕ 1 = 1**

**1 1 ⊕ 1 = 0**

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