John plans that, 'If I exercise and eat healthy, then I will feel great.' Which of the following best represents John's plan?

Understand the Problem

The question is asking which logical expression best represents John's plan regarding his exercise and healthy eating habits leading to feeling great. The options provided are different forms of logical statements.

Answer

$ (p \land q) \to r $

Steps to Solve

Identify the components of the statement

The original statement is: "If I exercise and eat healthy, then I will feel great."
Let:

$p$: "I exercise"
$q$: "I eat healthy"
$r$: "I feel great"

Formulate the logical expression

The statement can be expressed as a logical implication. It reads as:
$$ (p \land q) \to r $$
This means if both $p$ (I exercise) and $q$ (I eat healthy) are true, then $r$ (I feel great) is true.

Match with the answer choices

Now, let's check the options:

a. $p \to (p \land r)$
b. $(p \land q) \to r$
c. $p \lor (q \to r)$
d. $\neg p \lor q$

From our formulation, option b matches what we derived.

The best representation of John's plan is:
$ (p \land q) \to r $

More Information

This logical expression shows a conditional relationship where both exercising and eating healthy lead to feeling great. It's a standard way of expressing implications in logic.

Tips

Confusing the "and" ($\land$) with "or" ($\lor$), leading to an incorrect logical expression.
Overlooking how to correctly structure implications, thus selecting an option that doesn't represent the condition properly.

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