Logical Reasoning (statements, arguments)

Why A is correct:
The converse flips the hypothesis and conclusion. Original: "If even → then divisible by 2." Converse: "If divisible by 2 → then even."

Why the others are wrong:
- B is the contrapositive (flips AND negates both parts)—not the converse
- C is just a different statement; it doesn't follow the converse structure
- D restates the original statement exactly—no flip

A. The statement is true. ✓

• We're given that x = 2 is true, and we have a conditional: "If x = 2, then x² = 4."
• Since the condition is met (x = 2), we can deduce the conclusion: 2² = 4, which is mathematically correct.
• Deductive reasoning works by applying a true rule to a true premise, which gives us a true conclusion.

Why the others are wrong:
• B (False): The math checks out; 2² definitely equals 4.
• C (Partially true): Either a statement is true or it isn't—there's no partial truth here.
• D (Cannot be determined): We have all the information needed; this is straightforward deduction.

Why A is correct:
The premises are true (all primes are >1, and 7 is prime), so 7 is correctly determined to be >1. However, the conclusion claims 7 is >10, which doesn't logically follow from these premises. Valid arguments require the conclusion to follow necessarily from the premises—this one doesn't.

Why the others are wrong:
- B: Wrong because 7 is NOT greater than 10 (it's false), and even if it were true, that wouldn't make the argument valid. A valid argument needs correct logical structure, not just a true conclusion.
- C: Wrong because 7 IS prime, so this reason is factually incorrect.
- D: Wrong because an argument can have true premises and still be invalid if the conclusion doesn't logically follow from them.

A is correct. The contrapositive flips both parts of a statement AND negates both. Original: "If square → four equal sides." Contrapositive: "If NOT four equal sides → NOT square." This is logically equivalent to the original.

B is wrong. This flips the parts but doesn't negate correctly—it says "not square" in the condition when it should say "not four equal sides."

C is wrong. This just flips the original statement (the converse), which isn't the same as the contrapositive.

D is wrong. This also flips without proper negation and gets the logic backwards.

Correct Answer: A

The inverse of a conditional statement flips the truth value of both parts. The original statement is "If it rains (p), then the ground is wet (q)." The inverse is "If it does NOT rain (not p), then the ground will NOT be wet (not q)."

Why the others are wrong:
- B is the converse (swaps the parts but keeps them positive)
- C is the contrapositive (flips AND negates, which is actually logically equivalent to the original, not the inverse)
- D is just restating the original statement

A. Some is correct because some integers are divisible by 2 (like 2, 4, 6) while others are not (like 1, 3, 5). The word "some" means "at least one," which is true here.

B. All is wrong—not every integer is divisible by 2, since odd numbers aren't.

C. None is wrong—this would mean no integers are divisible by 2, which is false.

D. Most is wrong—while it might seem true, "most" is too strong and imprecise; exactly half of integers are divisible by 2, so "some" is the most accurate quantifier.

A is correct. The negation of "all" is "not all" (meaning at least one exception exists). This directly flips the original claim without changing its structure.

B is wrong. This says "all triangles lack three sides," which is too strong—it claims the opposite extreme rather than simply negating the universal statement.

C is wrong. This actually affirms a weaker version of the original claim rather than negating it; it doesn't contradict "all triangles have three sides."

D is wrong. This changes the subject from triangles to shapes, which doesn't negate the original statement at all.

A is correct: When combining statements with "and," both conditions must be true at the same time. This means the number must satisfy P (be even) AND satisfy Q (be greater than 10).

B is wrong: "Or" means at least one condition is true, not both. This changes the logical operator.

C is wrong: "Not even" contradicts statement P. We want the number to BE even, not NOT even.

D is wrong: "Less than 10" contradicts statement Q. We want the number to be greater than 10, not less than it.

Correct Answer: A. Inductive reasoning

You're drawing a general conclusion ("all even numbers are divisible by 2") from specific examples (the five even numbers you tested). That's inductive reasoning—moving from particular observations to a broad rule.

Why the others are wrong:
- B. Deductive reasoning goes the opposite direction: starting with a general rule and applying it to specific cases (e.g., "all even numbers are divisible by 2, so 8 is divisible by 2").
- C & D. Contrapositive and inverse reasoning are logical techniques for rewriting conditional statements, not about drawing conclusions from examples.

Why A (FALSE) is correct:
An implication P → Q is only false when P is true AND Q is false. Since we have exactly this situation, the implication is false.

Why the others are wrong:
- B (TRUE): Wrong because when the premise is true but the conclusion is false, the implication fails.
- C (Cannot be determined): Wrong because we're given specific truth values for both P and Q, so we can determine the answer with certainty.
- D (It depends): Wrong because the truth value of an implication is completely determined by the truth values of P and Q—there's nothing else it depends on.

A is correct. The converse flips the hypothesis and conclusion of an if-then statement. Original: "If study → then pass." Converse: "If pass → then study."

B is wrong. This is the contrapositive (flipping AND negating both parts), not the converse.

C is wrong. This contradicts the original statement and isn't a valid converse.

D is wrong. This negates the conclusion but doesn't flip the parts; it's neither a converse nor logically related.

A. TRUE is correct.

Using the Pythagorean theorem (a² + b² = c²), we can verify: 3² + 4² = 9 + 16 = 25 = 5². Since this equation holds true, the triangle IS a right triangle.

Why the others are wrong:
- B (FALSE) – The math proves the statement is correct, so it can't be false.
- C (Cannot be determined) – We have exact side lengths, so we can definitely determine this using the Pythagorean theorem.
- D (Partially true) – The statement is completely true for all triangles with sides 3, 4, and 5; there's no partial answer here.

Why A is correct:
The premises don't logically lead to the conclusion. Just because all cats are mammals AND some mammals are dogs doesn't mean all cats must be dogs—dogs could be a completely separate group of mammals. The argument breaks the rules of logical deduction.

Why B is wrong:
Being true that cats and dogs are both mammals doesn't make the argument valid. An argument is valid only if the conclusion *must* follow from the premises—and here it doesn't.

Why C is wrong:
Whether or not some cats are dogs is irrelevant. The problem isn't about what's actually true in the world; it's that the logical structure doesn't work.

Why D is wrong:
Again, what's factually true about mammals doesn't fix the logical flaw. The argument structure itself is faulty, so validity isn't saved by the premises being about real things.

# Explanation

Why A is correct:
The contrapositive flips AND negates both parts of an if-then statement. Original: "If rain → wet ground." Contrapositive: "If NOT wet ground → NOT rain." This is logically equivalent to the original statement.

Why the others are wrong:
- B reverses the statement without negating (this is the "converse"—not logically equivalent)
- C negates both parts without reversing them (this is the "inverse"—not logically equivalent)
- D keeps the same direction and negates only the conclusion (this is neither contrapositive, converse, nor inverse—it weakens the logic)

Correct Answer (A):
The inverse of a conditional statement negates both the hypothesis (if part) and conclusion (then part). Original: "If divisible by 5, then ends with 0 or 5." Inverse: "If NOT divisible by 5, then does NOT end with 0 or 5." ✓

Why others are wrong:
- B is the converse (swaps the if and then parts, not the inverse)
- C is the original statement unchanged
- D only negates part of the conclusion and doesn't negate the hypothesis, so it's neither inverse nor converse

Correct answer: Some

A polygon is a general shape category that includes triangles (3 sides), pentagons (5 sides), hexagons (6 sides), and many others—not all polygons have four sides. "Some" is correct because it accurately describes that certain polygons (quadrilaterals) do have four sides, without claiming this is true for every polygon.

Why the others are wrong:
- All – Incorrect; many polygons don't have four sides (like triangles)
- None – Incorrect; quadrilaterals are polygons with four sides, so at least some do
- Most – Too strong; we can't say most polygons have four sides (there are infinitely many with different side counts)

Correct Answer (A): "Not every integer is either odd or even."

To negate a statement starting with "every," you change it to "not every." This directly contradicts the original claim by saying the condition doesn't apply to all integers.

Why the others are wrong:

• B is too strong—it claims *no* integer is odd or even, which is false. We only need to say the original claim doesn't hold for all integers.
• C contradicts itself (nothing can be both odd AND even), so it's not a logical negation; it's nonsensical.
• D means the same thing as A, but uses awkward phrasing ("not all" vs. "not every"). While technically equivalent, A is the clearer, more standard way to negate a universal statement.

Why A is correct:
The question asks you to combine P and Q using "or" (the logical operator OR). "Or" means at least one statement is true. So you connect them: "The angle is acute OR the angle is less than 90 degrees."

Why others are wrong:
- B uses "and" instead of "or"—the question specifically asks for "or"
- C negates both statements and uses "or," which changes the original meaning completely
- D uses "and" and introduces "exactly 90 degrees," which contradicts both original statements (an angle can't be both acute and exactly 90°)

Correct Answer: Inductive Reasoning

You're drawing a general conclusion ("all swans are white") from specific observations (the white swans you've seen). That's inductive reasoning—moving from particular examples to a broad rule.

Why the others are wrong:
- Deductive reasoning works the opposite way: you start with a general rule and apply it to specific cases (e.g., "All swans are white, so this swan must be white").
- Contrapositive and inverse reasoning are logical techniques for reworking conditional statements, not about how you form conclusions from observations.

Why A (FALSE) is correct:
The conjunction operator ∧ (AND) only produces true when *both* P and Q are true. Since P is false, the entire statement P ∧ Q is false, regardless of Q's value.

Why the others are wrong:
- B (TRUE): Both parts would need to be true for AND to be true, but P is false.
- C (Cannot be determined): We have definite truth values for both P and Q, so we can absolutely determine the result.
- D (It depends): The result doesn't depend on anything else—it's determined by the fixed truth values given.