Question

Find the truth tables of the following propositions (1) \((\mathrm{A} \wedge \mathrm{B}) \vee(\mathrm{A} \wedge \mathrm{C})\) (2) \((\mathrm{A} \rightarrow \mathrm{B}) \wedge(\mathrm{B} \rightarrow \mathrm{C})\)

Short answer

The truth tables for the given propositions are: (1) For \((\mathrm{A} \wedge \mathrm{B}) \vee (\mathrm{A} \wedge \mathrm{C})\): A | B | C | (A AND B) OR (A AND C) ----|----|----|--------------------------- T | T | T | T T | T | F | T T | F | T | T T | F | F | F F | T | T | F F | T | F | F F | F | T | F F | F | F | F (2) For \((\mathrm{A} \rightarrow \mathrm{B}) \wedge (\mathrm{B} \rightarrow \mathrm{C})\): A | B | C | (A IMPLIES B) AND (B IMPLIES C) ----|----|----|------------------------------- T | T | T | T T | T | F | F T | F | T | F T | F | F | F F | T | T | T F | T | F | F F | F | T | T F | F | F | T

Step-by-step solution

Identify the sub-expressions and their truth tables

In this step, we identify the sub-expressions and their truth tables. In our case, we have two sub-expressions: \((\mathrm{A} \wedge \mathrm{B})\) and \((\mathrm{A} \wedge \mathrm{C})\). We start by listing all possible combinations of A, B, and C, and then find the truth values for these sub-expressions. Write down the truth tables for AND operation: A | B | A AND B ----|----|-------- T | T | T T | F | F F | T | F F | F | F A | C | A AND C ----|----|-------- T | T | T T | F | F F | T | F F | F | F

Combine the sub-expression truth tables

Now, we need to combine the truth tables for the sub-expressions using the OR operation. The OR operation gives "true" if any input is true. Write down the truth table for the entire expression. A | B | C | (A AND B) OR (A AND C) ----|----|----|--------------------------- T | T | T | T T | T | F | T T | F | T | T T | F | F | F F | T | T | F F | T | F | F F | F | T | F F | F | F | F (2) For the proposition \((\mathrm{A} \rightarrow \mathrm{B}) \wedge (\mathrm{B} \rightarrow \mathrm{C})\), we will again take it step by step.

Identify the sub-expressions and their truth tables

Here, we have two sub-expressions: \((\mathrm{A} \rightarrow \mathrm{B})\) and \((\mathrm{B} \rightarrow \mathrm{C})\). We start by listing all possible combinations of A, B, and C, and then find the truth values for these sub-expressions. Write down the truth tables for IMPLIES operation: A | B | A IMPLIES B ----|----|-------- T | T | T T | F | F F | T | T F | F | T B | C | B IMPLIES C ----|----|-------- T | T | T T | F | F F | T | T F | F | T

Combine the sub-expression truth tables

Now, we need to combine the truth tables using the AND operation. The AND operation gives "true" if both inputs are true. Write down the truth table for the entire expression. A | B | C | (A IMPLIES B) AND (B IMPLIES C) ----|----|----|------------------------------- T | T | T | T T | T | F | F T | F | T | F T | F | F | F F | T | T | T F | T | F | F F | F | T | T F | F | F | T

Key concepts

Logical Operators

Logical operators play a crucial role in propositional logic, which is a branch of discrete mathematics. They help in forming complex statements by combining simpler ones. The main logical operators used in truth tables include:

• AND (\(\land\)): This operator results in true only if both input statements are true. For example, \((A \land B)\) is true only if both \(A\) and \(B\) are true.


• OR (\(\lor\)): This operator is true if at least one of the input statements is true. So, \((A \lor B)\) is true if either \(A\) or \(B\) (or both) are true.


• IMPLIES (\(\rightarrow\)): This operator, also known as implication, is a bit tricky. \((A \rightarrow B)\) is false only when \(A\) is true and \(B\) is false, in all other cases it is true.

Understanding these operators helps in building and analyzing truth tables which are a systematic way to explore all possible truth values of a propositional expression. Truth tables are essential for determining the logical validity of an expression.

Propositional Logic

Propositional logic is the foundation of logical reasoning in mathematics and computer science. It involves statements that can be either true or false, known as propositions. In this logic system, we use propositions to build complex expressions using logical operators like AND, OR, and IMPLIES.Each proposition can be analyzed by using truth tables, which list every possible combination of truth values. They help determine the outcome of complex propositions across all these combinations.
For instance, consider the proposition \((A \wedge B) \vee (A \wedge C)\). This can be broken down to analyze the truth values of \((A \wedge B)\) and \((A \wedge C)\) separately, and then these results are combined using OR to derive the final truth values.Propositional logic is vital for logical reasoning because it forms the basis for constructing arguments and proofs. It provides a systematic way to assess the truth of complex statements based on their component parts.

Discrete Mathematics

Discrete mathematics is a branch of mathematics dealing with quantities that assume separate, distinct values. It is critical for computer science as it includes the study of algorithms, graphs, and logical reasoning. Within discrete mathematics, propositional logic and truth tables are key concepts. Logical operators and truth tables are used to evaluate and simplify logical expressions, making them crucial tools. This field equips students with the foundational knowledge needed to solve problems related to computer systems and software design. By mastering logical operators and propositional logic in discrete mathematics, you gain skills in precision and problem-solving, enhancing your ability to develop algorithms and analyze their behavior.
Discrete mathematics doesn't rely on continuity, unlike calculus, making it particularly suitable for codes and computations in computer technology. Understanding its core principles fosters the development of robust and efficient technological solutions.