To create a truth table for the expression $(p \rightarrow q) \vee (q \rightarrow r)$, we need to consider the truth values for all possible combinations of the variables $p$, $q$, and $r$.

Step-by-Step Breakdown:

1. Implication ($\rightarrow$): The statement $A \rightarrow B$ (if $A$, then $B$) is false only when $A$ is true and $B$ is false. In all other cases, it is true.
2. Disjunction ($\vee$): The statement $A \vee B$ (A or B) is true if at least one of $A$ or $B$ is true.

Let's break down the components of $(p \rightarrow q) \vee (q \rightarrow r)$ into a truth table:

$p$	$q$	$r$	$p \rightarrow q$	$q \rightarrow r$	$(p \rightarrow q) \vee (q \rightarrow r)$
T	T	T	T	T	T
T	T	F	T	F	T
T	F	T	F	T	T
T	F	F	F	T	T
F	T	T	T	T	T
F	T	F	T	F	T
F	F	T	T	T	T
F	F	F	T	T	T

Explanation:

1. Columns for $p$, $q$, and $r$: These list all possible truth values (True, False) for $p$, $q$, and $r$.
2. Column $p \rightarrow q$: True except when $p$ is True and $q$ is False.
3. Column $q \rightarrow r$: True except when $q$ is True and $r$ is False.
4. Final Column $(p \rightarrow q) \vee (q \rightarrow r)$: True when at least one of $p \rightarrow q$ or $q \rightarrow r$ is true.

Thus, the compound statement $(p \rightarrow q) \vee (q \rightarrow r)$ is always true for all possible values of $p$, $q$, and $r$.

Would you like more details on any of these steps, or have other questions?

Related Questions:

1. What are the truth tables for $p \rightarrow q$ and $q \rightarrow r$ individually?
2. How do truth tables relate to logical equivalences?
3. Can you simplify the expression $(p \rightarrow q) \vee (q \rightarrow r)$?
4. What is the significance of tautologies in logical statements?
5. How do you represent the logical disjunction ($\vee$) in different programming languages?

Tip: Understanding the properties of logical connectives like implication and disjunction can help in reducing complex logical statements to simpler forms.