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:
-
Original Statement (Conditional):
- "If is less than 7, then is less than or equal to 7."
- Symbolically: .
-
Contrapositive:
The contrapositive of a statement is , where denotes negation.
For :- : , so : .
- : , so : .
- Contrapositive: "If , then ."
-
Evaluate the Contrapositive:
The statement "If , then " is true because if is greater than 7, it is certainly greater than or equal to 7. -
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:
- What is the converse of the statement, and is it true?
- What is the inverse of the statement, and is it true?
- Can a conditional statement be true if its converse is false?
- How does negation apply to compound inequalities like ?
- 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
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