Problem 61 - Discrete Mathematics with Applications

Question

The Sheffer stroke / is a binary operator" defined by the following truth table.(Note: On page 25 we used the vertical bar \(|\) to mean is a factor of. The actual meaning should be clear from the context. So be careful.) Verify each. (Note: Exercise 58 shows that the logical operators \(|\) and \(\mathrm{NAND}\) are the same. (TABLE CAN'T COPY) $$p \rightarrow q \equiv((p | p)|(p | p))|(q | q)$$

Step-by-Step Solution

Answer

The given expression is equivalent to the Sheffer stroke (NAND) operator, as verified by the identical truth tables: *NAND:* *True | True | False* *True | False | True* *False | True | True* *False | False | True* *Expression:* *True | True | False* *True | False | True* *False | True | True* *False | False | True*

Step 1: Understand the Sheffer stroke (NAND) operator

The Sheffer stroke operator, also known as the NAND operator, is a binary operator that combines two logical statements (p and q), and returns "True" if and only if NOT both p and q are True. The truth table for the NAND operator is as follows: *p | q | NAND* *True | True | False* *True | False | True* *False | True | True* *False | False | True*

Step 2: Analyze the given expression

The given expression is \( p \rightarrow q \equiv((p | p)|(p | p))|(q | q) \) Now, we must analyze this expression and create a truth table for it:

Step 3: Create a truth table for the given expression

Create a truth table with columns for p, q, p|p, q|q, p|p ⊕ p|p, and (p|p ⊕ p|p) ⊕ (q|q): *p | q | p|p | q|q | (p|p)|(p|p) | ((p|p)|(p|p))|(q|q)* *True | True | True | True | False | False* *True | False | True | False | False | True* *False | True | False | True | True | True* *False | False | False | False | True | True*

Step 4: Compare the truth tables

Now that we have the truth table for the expression, let's compare this with the NAND operation truth table: *NAND:* *True | True | False* *True | False | True* *False | True | True* *False | False | True* *Expression:* *True | True | False* *True | False | True* *False | True | True* *False | False | True* Both truth tables are identical, meaning that the given expression is equivalent to the Sheffer stroke (NAND) operator. Therefore, we have successfully verified that \( p \rightarrow q \equiv((p | p)|(p | p))|(q | q) \).

Key Concepts

NAND Operator

The NAND operator, also known as the Sheffer stroke, is a fundamental logical operator in the realm of digital logic and mathematical logic. It stands for "Not AND" and operates on two binary inputs. It returns "True" unless both inputs are "True," in which case, it returns "False." This property makes NAND an interesting and versatile operator, as it is functionally complete, meaning you can express any logical operation such as AND, OR, or NOT using only NAND. The practical implementation of the NAND operator is vital due to its use in constructing digital circuits. In circuit design, NAND gates are often preferred because they can perform any Boolean function when used in the right configuration. This versatility and efficiency in operations make NAND indispensable in electronic devices.

Truth Table

A truth table is an essential tool used in logic to understand and depict how logical expressions work based on all possible input values. It lists all the potential combinations of inputs and the respective outputs of a given logical operation. For the Sheffer stroke or the NAND operator, creating a truth table helps reveal the behavior of this binary operator across its possible input states:

• True, True: Output is False.
• True, False: Output is True.
• False, True: Output is True.
• False, False: Output is True.

By examining these entries, one can grasp how the NAND operator functions and verify logical equivalences by comparing its truth table with another logical expression.

Logical Equivalence

Logical equivalence is a crucial concept in logic, particularly concerning propositions and their transformations. Two statements, propositions, or expressions are considered logically equivalent if they have the same truth value in every possible situation.In our exercise, the expression \( p \rightarrow q \equiv ((p | p)|(p | p))|(q | q) \) is logically equivalent to the logical implication "if \( p \) then \( q \)." By providing a truth table and showing that both the given expression and the implication yield the same results under all possible input conditions, we demonstrate their equivalence. Understanding logical equivalence allows you to simplify complex expressions and analyze logical statements more efficiently, maintaining correctness while reducing complexity.

Binary Operator

A binary operator is an operator that acts upon two operands. In logic, binary operators like AND, OR, and NAND work by taking two inputs and producing a single output based on their defined rules. The Sheffer stroke (NAND) is a prime example of a binary operator because it combines two inputs to determine the truth value according to its operation. Each binary operator has its unique set of rules and outcomes:

• AND: True if both operands are True.
• OR: True if at least one operand is True.
• NAND: True if not both operands are True.

Being proficient with binary operators is essential for fields involving logic expressions and digital computing, as they form the basic building blocks of more complex logical structures.