Math Problem Statement

is the contrapositive of the following conditional true? "if t is less than 7, then t is less than or equal to 7"

Solution

To determine if the contrapositive of the given conditional statement is true, let's proceed step by step:

  1. Original Statement (Conditional):

    • "If tt is less than 7, then tt is less than or equal to 7."
    • Symbolically: t<7    t7t < 7 \implies t \leq 7.
  2. Contrapositive:
    The contrapositive of a statement p    qp \implies q is ¬q    ¬p\neg q \implies \neg p, where ¬\neg denotes negation.
    For t<7    t7t < 7 \implies t \leq 7:

    • qq: t7t \leq 7, so ¬q\neg q: t>7t > 7.
    • pp: t<7t < 7, so ¬p\neg p: t7t \geq 7.
    • Contrapositive: "If t>7t > 7, then t7t \geq 7."
  3. Evaluate the Contrapositive:
    The statement "If t>7t > 7, then t7t \geq 7" is true because if tt is greater than 7, it is certainly greater than or equal to 7.

  4. Conclusion:
    The contrapositive is true, and since a conditional statement and its contrapositive are logically equivalent, the original statement is also true.

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Related Questions:

  1. What is the converse of the statement, and is it true?
  2. What is the inverse of the statement, and is it true?
  3. Can a conditional statement be true if its converse is false?
  4. How does negation apply to compound inequalities like a<bca < b \leq c?
  5. Is the logical equivalence between a statement and its contrapositive always guaranteed?

Tip:

The contrapositive of a conditional statement is always logically equivalent to the original statement. Use this fact to check validity efficiently!

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Math Problem Analysis

Mathematical Concepts

Logic
Conditional Statements
Contrapositive

Formulas

If p implies q (p → q), then the contrapositive is ¬q → ¬p

Theorems

Logical Equivalence between a Conditional Statement and its Contrapositive

Suitable Grade Level

Grades 9-12