The problem asks when the logical implication $p \rightarrow q$ is considered true. We are given 5 options and need to choose the correct one.

1. Problem Description

The problem asks when the logical implication $p \rightarrow q$ is considered true. We are given 5 options and need to choose the correct one.

2. Solution Steps

The implication $p \rightarrow q$ is read as "if $p$, then $q$". Its truth table is as follows:

| p | q | p -> q |

|---|---|--------|

| T | T | T |

| T | F | F |

| F | T | T |

| F | F | T |

From the truth table, we can see that $p \rightarrow q$ is true in all cases except when $p$ is true and $q$ is false.

Option (1) states "Only when both p and q are true". This is incorrect because $p \rightarrow q$ is also true when p is false.

Option (2) states "Only when both p and q are false". This is incorrect as well, because $p \rightarrow q$ is true when both are true, and when $p$ is false and $q$ is true.

Option (3) states "Provided p is false or q is true (or both)". This matches the truth table where $p \rightarrow q$ is true when p is false, q is true, or both.

Option (4) states "Provided p is true and q is true". This is incorrect because the implication is also true when p is false.

Option (5) states "Provided p is true or q is false (or both)". This is incorrect because when p is true and q is true, the implication should be true, but it doesn't meet this condition.

3. Final Answer

(3) Provided p is false or q is true (or both)