Question 1164609: Given the following logical expression, place the operations in the order in which you would apply, from first operation to last.
~p ∧ q → r
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! Following the standard rules of **Logical Precedence**, we evaluate the operations from the most "tightly bound" operator to the "main connective."
The hierarchy is as follows:
1. **Negation** ($\sim$)
2. **Conjunction** ($\land$)
3. **Disjunction** ($\lor$)
4. **Conditional** ($\to$)
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### **Order of Application**
For the expression **$\sim p \land q \to r$**, the operations should be applied in this order:
1. **First: Negation ($\sim p$)**
The "NOT" operator is applied only to the variable immediately following it. You must flip the truth value of $p$ first.
2. **Second: Conjunction ($\sim p \land q$)**
The "AND" operator takes precedence over the conditional. You evaluate the relationship between the result of Step 1 and $q$.
3. **Last: Conditional ($\dots \to r$)**
The "IF...THEN" operator is the main connective here. It evaluates the relationship between the entire left-hand side (the antecedent) and $r$ (the consequent).
### **Summary Table**
| Order | Operation | Symbol | Resulting Grouping |
| :--- | :--- | :--- | :--- |
| **1st** | Negation | $\sim$ | $(\sim p)$ |
| **2nd** | Conjunction | $\land$ | $(\sim p \land q)$ |
| **3rd** | Conditional | $\to$ | $((\sim p \land q) \to r)$ |
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**Wait, why not the arrow first?**
In logic, the conditional ($\to$) is considered "weaker" than conjunction or negation. If you wanted the arrow to be evaluated first, the expression would have required parentheses like this: $\sim p \land (q \to r)$. Without them, we follow the hierarchy above.
Are you working on a truth table for this specific statement?
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