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QUESTION 21
Create truth tables to show all possible inputs and outputs for the following Boolean functions:
- (A ˄B˄ ¬C)
- ¬ (¬ A ˅¬ B˅C)
Do not modify the Boolean functions in any way.
What similarities do you find when you compare them against one another?
Who proved the theorem regarding the properties you just uncovered?

QUESTION 22
If a Boolean function has five inputs (A, B, C, D, E), how many combinations/rows are required for its truth table?

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These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

First, we will introduce now the concept of logical variable. It is a variable whose values can be only logical values - True (usually denoted by T) and False (usually denoted by F or ). Therefore, it is ⊥ almost the same as the normal numerical variable, and the only difference is that it has logical values instead of numerical values (numbers).
Now, between the logical variables we can define logical operations. Similarly to numerical operations (like +, -, /, ...), there are also logical operations which take one (if unary) or two (if binary) logical variables and yield a result which is also a logical value.
The logical expression is any combination of logical variables and logical operations. For example, "P (Q ∨ ∧ ¬R)" is an expression where P, Q and R are logical variables which can take values T or F, and , , and ¬ are the logical operations. So, if we choose values for P, Q and R, we can evaluate ∨ ∧ every logical expression and the result will be again a logical value T or F.
To clearly describe the logical operations, we can use truth tables. For the truth table, we imagine a logical variable (or a few of them if needed), and then think what would be the result of the operation in each case (in case variable takes value T and in case variable takes value F).
First, we will consider the truth tables for the 5 basic logical operations. When we learn them, we will be able to solve more complex problems, with the combinations of different logical operations.
1. The first operation which we will discuss is called negation or logical NOT (denoted by symbol ¬). This operation simply negates the value - the negation of T is equal to F, and the negation of F is equal to T. The truth table for this operation...

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