How Do Truth Tables and Venn Diagrams Verify Logical Equivalence?

In summary, the conversation discusses how to use a truth table to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r). It also mentions designing a Venn diagram for this and proving that the statement is not equivalent to (p v q) ^ r. The conversation suggests filling out column 8 of the truth table and verifying that it matches column 5. This will provide evidence for the equivalence of the two statements. However, it is also important to note that the statement (p v q) ^ r is not equivalent to (p v q) ^ (p v r).
  • #1
barbara
10
0
How would I use a truth table to show that the statement p v (q ^ r) is equivalent to (p v q) ^ (p v r) or design a venn diagram for this. and show that this statement is not equivalent to (p v q) ^ r.
 
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  • #2
A truth-table might begin like this:

$\begin{array}{cccccccc}

P&Q& R&Q\wedge R&P\vee(Q\wedge R)& P\vee Q& P\vee R& (P\vee Q)\wedge(P\vee R)\\
\ast&\ast&\ast&\ast&\ast&\ast&\ast&\ast\\
T&T&T&T&T&T&T&-\\
T&T&F&F&T&T&T&-\\
T&F&T&F&T&T&T&-\\
T&F&F&F&T&T&T&-\\
F&T&T&T&T&T&T&-\\
F&T&F&F&F&T&F&-\\
F&F&T&F&F&F&T&-\\
F&F&F&F&F&F&F&-\\

\end{array}$

Your goal, then, is to fill out column 8, and verify it matches column 5...
 

Related to How Do Truth Tables and Venn Diagrams Verify Logical Equivalence?

1. What is a Venn Diagram?

A Venn Diagram is a visual representation of the relationships between different sets or groups. It consists of overlapping circles or shapes that show how the sets are related to each other.

2. How do you create a Venn Diagram?

To create a Venn Diagram, you first need to identify the sets or groups you want to compare. Then, draw circles or shapes representing each set, making sure that they overlap in the areas where the sets have common elements. Finally, label each section of the Venn Diagram with the corresponding elements or characteristics.

3. What is the purpose of a Venn Diagram?

The purpose of a Venn Diagram is to visually represent the relationships between different sets or groups. It can help to identify similarities, differences, and overlaps between the sets, making it easier to understand complex information and relationships.

4. What is a Truth Table?

A Truth Table is a mathematical table used to determine the truth values of a logical expression. It lists all possible combinations of truth values for the variables in the expression and shows the resulting truth value for the entire expression.

5. How do you use a Truth Table?

To use a Truth Table, you first need to identify the variables and the logical operators in the expression. Then, create a table with columns for each variable and the final expression. Fill in the rows with all possible combinations of true and false values for the variables, and use the logical operators to determine the resulting truth value for the expression in each row.

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