3The original is TRUE (anything above 5 is above 2), so the contrapositive is also TRUE.
✓ "If $x \leq 2$, then $x \leq 5$" — TRUE (logically equivalent to original)
Example 3. "If a shape is a square, then it is a rectangle." Which is logically equivalent to this? (A) If it is a rectangle, then it is a square. (B) If it is not a rectangle, then it is not a square.
2(A) is the Converse (Q → P) — NOT equivalent. (B) is ¬Q → ¬P — the Contrapositive.
✓ Answer: (B). The contrapositive is always logically equivalent.
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01
Forming the Contrapositive · Basic
Consider the statement: "If a number is divisible by 4, then it is divisible by 2." Which of the following is the contrapositive of this statement?
02
Days & Calendar · Basic
Given: "If today is Monday, then tomorrow is Tuesday." What is the contrapositive?
03
Algebra & Contrapositive · Basic
Statement: "If $x^2 = 9$, then $x = 3$." Which is the contrapositive of this statement?
04
Logical Equivalence · Basic
"If an animal is a dog, then it is a mammal." Which statement is logically equivalent to this?
05
Everyday Logic · Basic
"If I study hard, then I will pass the exam." Which is the contrapositive of this statement?
06
Number Theory · Intermediate
Consider the true statement: "If $n$ is an even integer, then $n^2$ is even." Which of the following is both the contrapositive AND true?
07
Geometry · Intermediate
"If a triangle has three equal sides, then all its angles measure 60°." Which is the correct contrapositive?
08
Science & Logic · Intermediate
"If it is snowing, then the temperature is below 0°C." Select the contrapositive:
09
Number Theory · Intermediate
"If $p$ is a prime number greater than 2, then $p$ is odd." What is the contrapositive of this statement?
10
Truth Value Analysis · Intermediate
Suppose the conditional statement $P \rightarrow Q$ is false. What can be concluded about its contrapositive $\neg Q \rightarrow \neg P$?
11
Double Negation · Intermediate
"If it is not raining, then I do not bring an umbrella." What is the contrapositive? (Simplify all double negations.)
12
Universal Statements · Intermediate
The statement "All cats are animals" can be written as the conditional: "If it is a cat, then it is an animal." Which is the contrapositive of this conditional?
13
Inequalities · Intermediate
The statement "If $x > 3$, then $2x > 6$" is true. Which of the following is the contrapositive, and what is its truth value?
14
Chained Conditionals · Advanced
Suppose $A \rightarrow B$ and $B \rightarrow C$ are both true. By hypothetical syllogism, $A \rightarrow C$ is true. What is the contrapositive of $A \rightarrow C$?
15
Proof by Contrapositive · Advanced
A student wants to prove: "If $n^2$ is odd, then $n$ is odd" using proof by contrapositive. Which statement should the student actually prove?
16
Biconditional · Advanced
The biconditional $P \leftrightarrow Q$ means $(P \rightarrow Q) \wedge (Q \rightarrow P)$. Taking the contrapositive of both component conditionals, which expression is logically equivalent to $P \leftrightarrow Q$?
17
Inference Rules · Advanced
Which argument form directly applies the contrapositive ($P \rightarrow Q \equiv \neg Q \rightarrow \neg P$)?
(A) From $P \rightarrow Q$ and $P$, conclude $Q$.
(B) From $P \rightarrow Q$ and $Q$, conclude $P$.
(C) From $P \rightarrow Q$ and $\neg Q$, conclude $\neg P$.
(D) From $P \rightarrow Q$ and $Q \rightarrow R$, conclude $P \rightarrow R$.
18
Truth Value Comparison · Advanced
The statement "If $x$ is a multiple of 6, then $x$ is a multiple of 3" is true. Which of the following related statements must also be true?
19
Compound Statements & De Morgan · Advanced
"If it is a weekend and it is sunny, then I go to the beach." Using De Morgan's Law ($\neg(A \wedge B) \equiv \neg A \vee \neg B$), what is the fully simplified contrapositive?
20
Set Theory · Challenge
Consider the statement: "If $A \subseteq B$, then $A \cap B = A$." ($A \subseteq B$ means every element of $A$ is also in $B$.)
Which of the following is the correct contrapositive?