Math Problem Statement

  1. The contrapositive of ”If x∈N,then x is integer and x>0"is-----------------------------------------------------------------------------------------------------------------------------------------

Solution

The contrapositive of the statement "If xNx \in \mathbb{N}, then xx is an integer and x>0x > 0" is:

"If xx is not an integer or x0x \leq 0, then xNx \notin \mathbb{N}."

Explanation:

  1. The original statement is of the form "If PP, then QQ," where:

    • PP: xNx \in \mathbb{N} (x belongs to the set of natural numbers)
    • QQ: xx is an integer and x>0x > 0
  2. The contrapositive of "If PP, then QQ" is "If not QQ, then not PP."

    • Not QQ: xx is not an integer or x0x \leq 0
    • Not PP: xNx \notin \mathbb{N} (x does not belong to the set of natural numbers)

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Further Questions:

  1. What is the difference between the contrapositive, converse, and inverse of a statement?
  2. How do you prove that the contrapositive of a statement is logically equivalent to the original statement?
  3. What are some examples of contrapositive statements in real-life scenarios?
  4. How does the concept of contrapositive help in mathematical proofs, especially in proving implications?
  5. Can you explain the role of contrapositive in logical reasoning and decision-making processes?

Tip:

When constructing the contrapositive, remember that you negate both parts of the statement and reverse the direction of the implication.