The following exercises involve the logical operators N A N...

The following exercises involve the logical operators $N A N D$ and $N O R$. The proposition $p$ NAND $q$ is true when either $p$ or $q$, or both, are false; and it is false when both $p$ and $q$ are true. The proposition $p$ NOR $q$ is true when both $p$ and $q$ are false, and it is false otherwise. The propositions $p$ NAND $q$ and $p$ NOR $q$ are denoted by $p \mid q$ and $p \downarrow q$, respectively. (The operators | and $\downarrow$ are called the Sheffer stroke and the Peirce arrow after H. M. Sheffer and C. S. Peirce, respectively.) Construct a truth table for the logical operator $\mathrm{NOR}$.

The following exercises involve the logical operators $N A N D$ and $N O R$. The proposition $p$ NAND $q$ is true when either $p$ or $q$, or both, are false; and it is false when both $p$ and $q$ are true. The proposition $p$ NOR $q$ is true when both $p$ and $q$ are false, and it is false otherwise. The propositions $p$ NAND $q$ and $p$ NOR $q$ are denoted by $p \mid q$ and $p \downarrow q$, respectively. (The operators | and $\downarrow$ are called the Sheffer stroke and the Peirce arrow after H. M. Sheffer and C. S. Peirce, respectively.)
Construct a truth table for the logical operator $\mathrm{NOR}$.

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Key Concepts

Logical Operators

Logical operators are symbols that represent operations on logical values or propositions. They are the foundational elements in Boolean algebra that allow us to build and analyze complex logical statements by combining simpler propositions.

Truth Table

A truth table is a systematic tabulation of all possible truth values for given logical variables and the resulting truth of a compound statement for each combination. It is an essential tool in logic that helps in understanding and verifying the behavior of logical expressions.

NOR Operator

The NOR operator is a logical connective that outputs true only when all its input propositions are false, and false otherwise. It is defined as the negation of the OR operator, making it a key function in Boolean logic for creating logical complements.

Alternative Notations (Sheffer Stroke and Peirce Arrow)

The sheffer stroke and the Peirce arrow are alternative symbols used for NAND and NOR operations respectively. Named after logicians H. M. Sheffer and C. S. Peirce, these notations facilitate the study and implementation of logical operations in theoretical and applied contexts.

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