12 multiple-choice questions, progressively harder.
Which example refutes “Every angle smaller than a straight angle is acute”?
Solution
Correct answer: C
The hypothesis requires a measure smaller than 180 degrees.
An angle of 124 degrees satisfies that condition but is obtuse, not acute. The 180-degree and 250-degree angles do not satisfy the hypothesis.
A conditional is true. Its converse is false. Which statement must also be false?
Correct answer: A
The inverse is logically equivalent to the converse.
They therefore have the same truth value, while the contrapositive matches the original.
Circle CCC has center OOO and positive radius rrr. Let ppp mean “the point is on CCC” and qqq mean “its distance from OOO is rrr.” Which formula is the contrapositive of p→qp\to qp→q?
Correct answer: D
Reverse the order and negate both statements.
The result is ¬q→¬p.\neg q\to\neg p.¬q→¬p.
The claim is “For every point M, if AM = MB, then M is the midpoint of segment AB.” Which missing condition is needed?
Correct answer: B
A midpoint must lie on the segment as well as divide it into equal lengths.
Equality of distances alone does not give the location condition.
All rectangles are quadrilaterals. A figure is a quadrilateral. What can be concluded about whether it is a rectangle?
The given fact is the conclusion of the conditional, not its hypothesis.
Both rectangles and other quadrilaterals fit the information. Reversing the implication is not licensed.
A claimed counterexample has a false hypothesis and a false conclusion. Does it refute the conditional?
A conditional only promises its conclusion when its hypothesis holds.
To refute it, a counterexample must have a true hypothesis and a false conclusion.
A point P is on a circle with center O and radius 12 cm. Which inference uses the circle definition?
A circle is the set of all points in a plane at the stated positive distance from its center in that plane.
Membership of P therefore gives OP = 12 cm. This is an exact deduction from the given, rather than a measurement.
Which biconditional is true under the lesson definitions?
Both directions of the correct statement reproduce the square definition.
The other statements drop a required condition in their reverse direction.
Every square is a rectangle. Which additional fact allows detachment to conclude that figure T is a rectangle?
Detachment needs the conditional and a fact matching its hypothesis.
The hypothesis is that the particular figure is a square.
Which pair of conditionals can be connected by syllogism to obtain “If p, then r”?
The conclusion of the first link must match the hypothesis of the next.
The premises are p→qp\to qp→q and q→rq\to rq→r. Syllogism gives the conclusion p→rp\to rp→r.
An exact construction is justified because two drawn segments are radii of the same compass circle. What kind of support is this?
The fixed compass opening makes both distances equal to the radius.
That exact reason applies independently of the accuracy of a printed ruler reading.
A square-to-rectangle conditional and its contrapositive are both true. What does this establish about the converse?
The contrapositive restates the same logical promise.
It adds no reverse implication. In this particular square-and-rectangle example, a nonsquare rectangle refutes the converse.
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