Math Problem Statement

If the water is not cold, then you cannot wear a wetsuit. Let W represent 'The water is cold' and Z represent 'You can wear a wetsuit'. Write the statement in symbolic form.

Solution

The problem asks you to express a logical statement in symbolic form based on the following conditions:

  • The sentence is: "If the water is not cold, then you cannot wear a wetsuit."
  • The given symbolic representations are:
    • WW: The water is cold.
    • ZZ: You can wear a wetsuit.

The logical structure of this statement is an implication. Breaking it down:

  • "The water is not cold" translates to ¬W\neg W (negation of WW).
  • "You cannot wear a wetsuit" translates to ¬Z\neg Z (negation of ZZ).

So, the full sentence becomes:

  • "If ¬W\neg W, then ¬Z\neg Z" or in symbolic form: ¬W¬Z\neg W \rightarrow \neg Z.

Now, looking at the options in the image, the correct symbolic form is the first one: ¬W¬Z\neg W \rightarrow \neg Z

Let me know if you need further clarification or have any questions!


Here are five related questions to enhance your understanding:

  1. What is the difference between ¬W¬Z\neg W \rightarrow \neg Z and WZW \rightarrow Z?
  2. How would you write the statement "If you can wear a wetsuit, then the water is cold" in symbolic form?
  3. What is the converse of the statement ¬W¬Z\neg W \rightarrow \neg Z?
  4. How would you express "You can wear a wetsuit if and only if the water is cold" using symbolic logic?
  5. What happens logically if both WW and ZZ are false in ¬W¬Z\neg W \rightarrow \neg Z?

Tip: In logic, implications can be tricky because PQP \rightarrow Q means "if PP is true, then QQ must be true," but it doesn't necessarily work in reverse!

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Math Problem Analysis

Mathematical Concepts

Logic
Propositional Logic
Implication

Formulas

Implication: P → Q

Theorems

Implication Theorem

Suitable Grade Level

College level introductory logic or discrete mathematics