Math Problem Statement

Let p, q, and r represent the following simple statements: p: It is snowing outside. q: It is cold. r: It is cloudy. Write the following compound statement in its symbolic form: 'If it is not snowing outside then: it is not cold and it is not cloudy.'

Solution

The given task asks for the symbolic representation of a compound statement based on the definitions of pp, qq, and rr:

  • pp: It is snowing outside.
  • qq: It is cold.
  • rr: It is cloudy.

The compound statement is:
"If it is not snowing outside, then it is not cold and it is not cloudy."

Solution:

  1. "If it is not snowing outside" translates to ¬p\neg p.
  2. "It is not cold" translates to ¬q\neg q.
  3. "It is not cloudy" translates to ¬r\neg r.
  4. "It is not cold and it is not cloudy" is ¬q¬r\neg q \land \neg r.
  5. The entire compound statement, using the "if-then" logic (\rightarrow), becomes:
    ¬p(¬q¬r)\neg p \rightarrow (\neg q \land \neg r)

Final Answer:

The symbolic form is:
¬p(¬q¬r)\neg p \rightarrow (\neg q \land \neg r)

Let me know if you need more clarification or a deeper explanation!

Related Questions:

  1. What does \rightarrow mean in logic, and how does it work?
  2. Can the statement ¬p(¬q¬r)\neg p \rightarrow (\neg q \land \neg r) be written in a different but equivalent form?
  3. What is the truth table for this compound statement?
  4. How does the "and" (\land) operator function when combined with negation?
  5. How can we convert compound statements into their English equivalents?

Tip:

Always remember that in logic, ¬\neg (negation) reverses the truth value of a statement, and \rightarrow means "if-then," where the statement is false only when the first part is true and the second part is false.

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Logic
Compound Statements

Formulas

Implication: p → q
Negation: ¬p
Conjunction: q ∧ r

Theorems

Rules of Logical Implication
De Morgan's Laws

Suitable Grade Level

Grades 9-12