Math Problem Statement

Consider the hierarchy of operators as: ¬,∧,∨,→,↔ And Give full parenthesization for the following propositions a. 𝑝→𝑝∨𝑞→𝑞 b. 𝑝↔𝑞→𝑟↔𝑝 c. 𝑝∨𝑞∧(𝑟∨𝑝)∧𝑟

Solution

In logic, the hierarchy of operators typically follows this order, from highest to lowest precedence:

  1. ¬ (negation)
  2. (conjunction)
  3. (disjunction)
  4. (implication)
  5. (biconditional)

Now, let's give the full parenthesization for the propositions provided, using the operator hierarchy.

a. ppqqp \to p \vee q \to q

This expression has implications and disjunctions. We will parenthesize according to the precedence.

  • First, ppp \to p is handled.
  • Then, ppqp \to p \vee q is evaluated (since \to has lower precedence than \vee).
  • Finally, the q\to q operation applies to the result of the previous.

Thus, the full parenthesization is:

(p(p(qq)))(p \to (p \vee (q \to q)))

b. pqrpp \leftrightarrow q \to r \leftrightarrow p

This expression involves biconditionals and implications. We follow the precedence order:

  • First, pqp \leftrightarrow q is evaluated.
  • Then, the implication r\to r is applied.
  • Finally, p\leftrightarrow p is applied to the result of the previous operations.

Thus, the full parenthesization is:

((pq)(rp))((p \leftrightarrow q) \to (r \leftrightarrow p))

c. pq(rp)rp \vee q \land (r \vee p) \land r

This expression involves conjunctions, disjunctions, and parentheses. The order of operations should respect the following hierarchy:

  • First, q(rp)q \land (r \vee p) is evaluated (because \land has higher precedence than \vee).
  • Then, the entire expression p(q(rp))p \vee (q \land (r \vee p)) is considered, followed by the last r\land r.

Thus, the full parenthesization is:

p(q(rp))rp \vee (q \land (r \vee p)) \land r


Summary of Full Parenthesizations:

a. (p(p(qq)))(p \to (p \vee (q \to q)))

b. ((pq)(rp))((p \leftrightarrow q) \to (r \leftrightarrow p))

c. p(q(rp))rp \vee (q \land (r \vee p)) \land r


Would you like more details on the step-by-step breakdown, or do you have any specific questions about the hierarchy of operators?

Here are some related questions you might find interesting:

  1. How does negation interact with other logical operators in terms of precedence?
  2. Can you explain the difference between implication (\to) and biconditional (\leftrightarrow) in detail?
  3. How do you simplify complex logical expressions step by step?
  4. What happens if we change the order of parentheses in a logical expression?
  5. Can you show how truth tables are constructed for these kinds of logical operations?

Tip: Always remember the hierarchy when dealing with complex logical expressions—it will guide you in determining the correct order of operations!

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Logic
Operator Precedence

Formulas

¬ (negation)
∧ (conjunction)
∨ (disjunction)
→ (implication)
↔ (biconditional)

Theorems

Hierarchy of logical operators

Suitable Grade Level

Grades 11-12