Question

Give truth tables for the following expressions. a. \((s \vee t) \wedge(\neg s \vee t) \wedge(s \vee \neg t)\) b. \((s \Rightarrow t) \wedge(t \Rightarrow u)\) c. \((s \vee t \vee u) \wedge(s \vee \neg t \vee u)\)

Short answer

Construct truth tables for each expression by evaluating all combinations of truth values for variables and compute results for sub-expressions to find the final outputs.

Step-by-step solution

Variables and Operations Identification

For each logical expression, identify the individual logical operations such as OR (\(\vee\)), AND (\(\wedge\)), NOT (\(eg\)), and IMPLICATION (\(\Rightarrow\)). Identify all variables involved. For expression (a) and (c), the variables are \(s\), \(t\), and \(u\). For expression (b), the variables are \(s\), \(t\), and \(u\).

Truth Table Setup

For each expression's truth table, list all possible truth values combinations for the involved variables. If there are three variables (\(s\), \(t\), \(u\)), you will have \(2^3=8\) rows for truth value combinations. If only \(s\) and \(t\) are involved, you will have \(2^2=4\) rows.

Compute Sub-Expressions

For each row in the truth table, compute any intermediate sub-expression results. This involves evaluating expressions under the operations \(eg\), \(\vee\), \(\wedge\), and \(\Rightarrow\). Write these intermediate results next to each row in the table. For example, compute \(eg s\), \(s \vee t\) before getting the final output in each row.

Calculate the Final Expression

Combine the results of the sub-expressions using \(\wedge\) and \(\Rightarrow\) as necessary to compute the truth value of the entire logical expression for each row. For example, combine \((s \vee t)\) with \((eg s \vee t)\) using \(\wedge\) to find the result of \((s \vee t) \wedge (eg s \vee t)\).

Record the Truth Table Results

Once you have calculated the final truth value of the expression for each row, fill out the column in the truth table that corresponds to the entire expression. Repeat these steps for expressions (b) and (c) by carefully analyzing their intermediate steps and finding the final outputs.

Key concepts

Logical Expressions

A logical expression is a combination of symbols and operators that represent a precise condition or statement in logic. These expressions can evaluate to either true or false based on the values inputted into them. For example, in the expressions provided in your exercise, we see heavy usage of operators such as "OR" (\(\vee\)), "AND" (\(\wedge\)), and others that connect variables. These variables, like \(s\), \(t\), and \(u\), are placeholders for either true or false values.

To understand logical expressions, it's important to break them down. For instance, the expression \((s \vee t) \wedge (eg s \vee t) \wedge (s \vee eg t)\) means that each part of the expression evaluates to true before the whole expression is true. Breaking expressions into their component parts is crucial for building truth tables where each possible truth combination is systematically evaluated.

Logical Operations

Logical operations are the rules that determine how combinations of truth values and expressions are transformed. The basic logical operations include "OR" (\(\vee\)), "AND" (\(\wedge\)), "NOT" (\(eg\)), and "IMPLICATION" (\(\Rightarrow\)). These operations allow us to create complex logical expressions used in calculations and decision-making processes.

• OR (\(\vee\)): This operation evaluates to true if at least one of the operands is true. For example, \(s \vee t\) is true if either \(s\) is true, \(t\) is true, or both are true.
• AND (\(\wedge\)): This operation evaluates to true only if both operands are true. For \(s \wedge t\), both \(s\) and \(t\) must be true for the expression to evaluate as true.
• NOT (\(eg\)): This unary operation inverts the truth value of its operand. If \(s\) is true, \(eg s\) (not \(s\)) is false, and vice versa.
• IMPLICATION (\(\Rightarrow\)): This operation evaluates to false only if the first operand is true and the second operand is false, otherwise it is true.

Understanding how these operations interact is critical for solving and constructing truth tables for logical expressions.

Truth Values

Truth values are the basic data types in logic. They denote whether a statement is "true" or "false," commonly represented as 1 for true and 0 for false. Evaluating a logical expression involves computing its truth value based on the combinations of truth values of its variables.

For each logical expression, you list all potential combinations of truth values for the variables involved. For example, if there are three variables \(s\), \(t\), and \(u\), you have a total of \(2^3 = 8\) combinations. Each row in the truth table corresponds to a different set of truth values for \(s\), \(t\), and \(u\), and guides you in computing the truth value of the expressions as part of solving logical problems.

Truth tables are systematic and help visually represent how different operations apply across different truth values, which aids in understanding and verifying logical expressions.

Boolean Algebra

Boolean Algebra is the mathematical framework that allows us to work with truth values and logical operators. It's a branch of algebra that deals with true or false values and is foundational in digital circuit design, computer science, and logic. Here, logical operations function similarly to algebraic operations.

In Boolean Algebra, we use specific laws and rules to simplify and evaluate logical expressions. Key laws include the law of identity, the law of null, the law of complementarity, and De Morgan's laws. These rules make it easier to combine, rearrange, and reduce logical expressions efficiently.

• Identity Law: \(a \vee 0 = a\) and \(a \wedge 1 = a\)
• Null Law: \(a \vee 1 = 1\) and \(a \wedge 0 = 0\)
• Complement Law: \(a \vee eg a = 1\) and \(a \wedge eg a = 0\)
• De Morgan’s Laws: \(eg(a \wedge b) = eg a \vee eg b\) and \(eg(a \vee b) = eg a \wedge eg b\)

Mastering Boolean Algebra is essential for simplifying complex logical expressions and for creating clear and accurate truth tables.