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Math 30-2 Diploma study materials
Math 30-2 Diploma study resources

Math 30-2 Diploma practice questions

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Question 1Logical Reasoning

Which statement is a proposition in the logical sense?

  1. Close the door, please.
  2. What time is it?
  3. The number 7 is odd. Correct answer.
  4. Let x equal 5.

Explanation

A proposition is a declarative statement that is either true or false. 'The number 7 is odd' makes a truth claim and is true. Commands, questions, and definitions are not truth-valued.

Question 2Logical Reasoning

The statement 'If it rains, then the ground is wet' is false only when:

  1. it rains and the ground is wet
  2. it does not rain and the ground is wet
  3. it rains and the ground is dry Correct answer.
  4. it does not rain and the ground is dry

Explanation

A conditional is false only when the hypothesis is true and the conclusion is false. So the statement fails only in the case where it rains but the ground stays dry. All other combinations make the conditional true.

Question 3Logical Reasoning

Consider the statement 'All dogs are mammals.' Which is its converse?

  1. All mammals are dogs. Correct answer.
  2. Some dogs are not mammals.
  3. All non-mammals are non-dogs.
  4. No dogs are mammals.

Explanation

The converse swaps the hypothesis and conclusion of a statement. Swapping 'all dogs are mammals' gives 'all mammals are dogs.' The converse of a true statement is not automatically true.

Question 4Logical Reasoning

Which pair of statements are negations of each other?

  1. 'It is cold' and 'It is cold or windy'
  2. 'It is cold' and 'It is freezing'
  3. 'It is cold' and 'It is not cold' Correct answer.
  4. 'It is cold' and 'It is windy'

Explanation

Two statements are negations when one is true exactly when the other is false. 'It is not cold' directly denies 'it is cold,' so they always have opposite truth values.

Question 5Logical Reasoning

In a truth table, how many rows are needed for a compound statement built from three distinct simple statements?

  1. 3
  2. 6
  3. 9
  4. 8 Correct answer.

Explanation

Each simple statement has 2 truth values, so n statements require 2^n rows. For three statements that is 2^3 = 8 rows.

Question 6Logical Reasoning

The statement 'Some students play chess' is false. What must be true?

  1. All students play chess.
  2. Most students play chess.
  3. No students play chess. Correct answer.
  4. Some students do not play chess.

Explanation

The negation of an existential claim ('some A are B') is a universal denial ('no A are B'). If it is false that at least one student plays chess, then no student does.

Question 7Logical Reasoning

A statement and its contrapositive always have which relationship?

  1. They are equivalent only when the original is true.
  2. They are always logically equivalent. Correct answer.
  3. They are always opposite in truth value.
  4. They are unrelated in truth value.

Explanation

A conditional and its contrapositive are logically equivalent: they have identical truth tables in every case. This is why proving the contrapositive proves the original.

Question 8Logical Reasoning

The statement 'If a shape is a square, then it has four equal sides' is true. Which related statement must also be true?

  1. If a shape is a square, then it does not have four equal sides.
  2. If a shape does not have four equal sides, then it is not a square. Correct answer.
  3. If a shape is not a square, then it does not have four equal sides.
  4. If a shape has four equal sides, then it is a square.

Explanation

Only the contrapositive is guaranteed true whenever the original is true. Negating and swapping gives 'if a shape does not have four equal sides, then it is not a square.'

Question 9Logical Reasoning

The compound statement 'P and Q' is true in exactly how many of the four possible cases of P and Q?

  1. Three
  2. Two
  3. Four
  4. One Correct answer.

Explanation

A conjunction is true only when both parts are true, which is one of the four cases. Every other combination makes at least one part false.

Question 10Logical Reasoning

'P or Q' is used in the inclusive sense. In which case is it false?

  1. P is false and Q is false Correct answer.
  2. P is true and Q is true
  3. P is false and Q is true
  4. P is true and Q is false

Explanation

An inclusive disjunction is true whenever at least one part is true. It is false only when both parts are false.

Question 11Logical Reasoning

A Venn diagram shows all cats inside the set of animals and all animals inside the set of living things. What must be true?

  1. Some animals are not living things.
  2. No living things are animals.
  3. All cats are living things. Correct answer.
  4. All living things are cats.

Explanation

Nested regions express a chain of universal statements: cats are animals and animals are living things. Transitivity gives that all cats are living things.

Question 12Logical Reasoning

A puzzle states: 'If Ana studies, she passes. Ana did not pass.' What logically follows?

  1. Ana passed anyway.
  2. Nothing can be concluded.
  3. Ana studied.
  4. Ana did not study. Correct answer.

Explanation

This is a valid use of modus tollens: from 'if P then Q' and 'not Q', conclude 'not P'. Since she did not pass, she did not study.

Question 13Logical Reasoning

'If Ben exercises, he is healthy. Ben is healthy. Therefore Ben exercises.' This reasoning is:

  1. valid by modus ponens
  2. invalid because it denies the antecedent
  3. valid by modus tollens
  4. invalid because it affirms the consequent Correct answer.

Explanation

The argument takes the conclusion of a conditional as a premise and infers the hypothesis, which is affirming the consequent. Other causes could make Ben healthy without exercise, so the form is invalid.

Question 14Logical Reasoning

A survey finds that every student who joined the chess club also joined the math club, and some math club members joined the chess club. Which statement is consistent with the data?

  1. Some chess club members are not in the math club.
  2. No chess club members are in the math club.
  3. All math club members are chess club members.
  4. Every chess club member is in the math club. Correct answer.

Explanation

The first finding states directly that all chess club members are also math club members. The second finding is compatible with this but does not require the reverse inclusion.

Question 15Logical Reasoning

Which statement is the biconditional 'P if and only if Q' true?

  1. Only when both P and Q are true
  2. When P and Q have opposite truth values
  3. When P and Q have the same truth value Correct answer.
  4. Only when both P and Q are false

Explanation

A biconditional asserts that the two parts always agree. It is true when both are true or both are false, and false otherwise.

Question 16Logical Reasoning

The statement 'If the light is green, then cars may go' is true. A car is not allowed to go. What can be concluded?

  1. The light is not green. Correct answer.
  2. The light is yellow.
  3. The light is green.
  4. Nothing can be concluded.

Explanation

Using modus tollens with the conditional and the denial of the conclusion gives the denial of the hypothesis. If cars may not go, the light is not green.

Question 17Logical Reasoning

A truth table shows a compound statement is true in every row. Such a statement is called:

  1. a tautology Correct answer.
  2. a contingency
  3. a contradiction
  4. a negation

Explanation

A tautology is a compound statement that is true under every possible assignment of truth values. A contradiction is false in every row, and a contingency is true in some rows and false in others.

Question 18Logical Reasoning

The statements 'If it is a rose, then it is a flower' and 'If it is not a flower, then it is not a rose' are:

  1. inverses of each other
  2. converses of each other
  3. unrelated statements
  4. contrapositives of each other Correct answer.

Explanation

The second statement negates and swaps the parts of the first, which is exactly the contrapositive. Contrapositives are logically equivalent to their originals.

Question 19Logical Reasoning

In a group of 30 students, 18 play soccer, 15 play basketball, and 8 play both. How many play neither sport?

  1. 7
  2. 3
  3. 5 Correct answer.
  4. 12

Explanation

By inclusion-exclusion, the number playing at least one sport is 18 + 15 - 8 = 25. Subtracting from 30 leaves 5 who play neither.

Question 20Logical Reasoning

A statement of the form 'All A are B' is false. Which situation makes it false?

  1. No A is B
  2. All B are A
  3. At least one B is not A
  4. At least one A is not B Correct answer.

Explanation

A universal affirmative is false exactly when a counterexample exists: some A that is not B. This single case refutes the claim.

Question 21Logical Reasoning

Two statements are logically equivalent. Which truth-table feature must they share?

  1. Their truth columns must match row for row. Correct answer.
  2. They must both be tautologies.
  3. They must use the same connectives.
  4. They must contain the same number of words.

Explanation

Logical equivalence means the two statements have identical truth values in every row. Word count, connectives, and tautology status are irrelevant to equivalence.

Question 22Logical Reasoning

'If the team wins, they celebrate. If they celebrate, they rest.' The team does not rest. What follows?

  1. The team does not win. Correct answer.
  2. The team wins.
  3. The team rests anyway.
  4. The team celebrates.

Explanation

Chaining the conditionals gives 'if the team wins, they rest.' With 'they do not rest,' modus tollens yields that the team does not win.

Question 23Logical Reasoning

A statement's inverse is true. What can be concluded about the original statement?

  1. Nothing definite about the original. Correct answer.
  2. The original is equivalent to its converse.
  3. The original must be true.
  4. The original must be false.

Explanation

The inverse is equivalent to the converse, not to the original. A true inverse therefore tells you nothing certain about the original statement's truth value.

Question 24Logical Reasoning

A compound statement uses 'or' to join two statements that can never both be true. Which connective best matches everyday intent here?

  1. Exclusive or Correct answer.
  2. Inclusive or
  3. Biconditional
  4. Conjunction

Explanation

When two options cannot both hold, everyday 'or' means exclusive or: exactly one is true. Inclusive or would allow both, which the situation rules out.

Question 25Logical Reasoning

Which statement is the negation of 'All prime numbers are odd'?

  1. Some prime numbers are not odd. Correct answer.
  2. All prime numbers are even.
  3. No prime numbers are odd.
  4. Some odd numbers are not prime.

Explanation

The negation of a universal affirmative is an existential claim naming a counterexample. 'Some prime numbers are not odd' captures that exception, and 2 is indeed not odd.

Question 26Logical Reasoning

Which of the following is a statement?

  1. The sky is blue. Correct answer.
  2. Wow, that's amazing!
  3. What time is it?
  4. Close the door.

Explanation

A statement is a declarative sentence that is either true or false. Questions, commands, and exclamations are not statements because they do not have a truth value.

Question 27Logical Reasoning

What is the negation of the statement 'Some birds can swim'?

  1. Some birds cannot swim.
  2. All birds can swim.
  3. No birds can swim. Correct answer.
  4. Not all birds can swim.

Explanation

The negation of 'Some A are B' is 'No A are B'. It is not enough to say that some A are not B, because both 'some are' and 'some are not' can be true simultaneously.

Question 28Logical Reasoning

The statement 'If it rains, then the ground is wet' is true. Which of the following must also be true?

  1. If it rains, then the ground is not wet.
  2. If the ground is wet, then it rains.
  3. If the ground is not wet, then it does not rain. Correct answer.
  4. If it does not rain, then the ground is not wet.

Explanation

The contrapositive of a true conditional statement is always true. The contrapositive of 'If P then Q' is 'If not Q then not P'.

Question 29Logical Reasoning

The statement 'All dogs are mammals' is given. What is its converse?

  1. Some dogs are not mammals.
  2. All mammals are dogs. Correct answer.
  3. No dogs are mammals.
  4. If an animal is not a mammal, then it is not a dog.

Explanation

The converse of 'All A are B' is 'All B are A'. Here, A is dogs and B is mammals, so the converse is 'All mammals are dogs'.

Question 30Logical Reasoning

Which of the following is the inverse of the statement 'If a figure is a square, then it has four sides'?

  1. If a figure is not a square, then it does not have four sides. Correct answer.
  2. If a figure has four sides, then it is a square.
  3. If a figure is a square, then it does not have four sides.
  4. If a figure does not have four sides, then it is not a square.

Explanation

The inverse of 'If P then Q' is 'If not P then not Q'. Here P is 'a figure is a square' and Q is 'it has four sides'.

Question 31Logical Reasoning

The statement 'If x > 3, then x > 1' is true. Which related statement must also be true?

  1. If x ≤ 1, then x ≤ 3. Correct answer.
  2. If x ≤ 3, then x ≤ 1.
  3. If x > 3, then x ≤ 1.
  4. If x > 1, then x > 3.

Explanation

The contrapositive of a true conditional is always true. The contrapositive of 'If P then Q' is 'If not Q then not P'. Here, not Q is x ≤ 1 and not P is x ≤ 3.

Question 32Logical Reasoning

The compound statement 'P and Q' is false. Which of the following must be true?

  1. P is true and Q is false.
  2. At least one of P or Q is false. Correct answer.
  3. Both P and Q are false.
  4. P is false and Q is true.

Explanation

A conjunction 'P and Q' is true only when both P and Q are true. Therefore, if it is false, at least one of P or Q must be false.

Question 33Logical Reasoning

The statement 'P or Q' is true. Which of the following must be true?

  1. Q is true.
  2. Both P and Q are true.
  3. At least one of P or Q is true. Correct answer.
  4. P is true.

Explanation

An inclusive disjunction 'P or Q' is true if at least one of P or Q is true. It does not require both to be true.

Question 34Logical Reasoning

Which of the following is the negation of 'All students like math'?

  1. Some students do not like math. Correct answer.
  2. All students do not like math.
  3. Some students like math.
  4. No students like math.

Explanation

The negation of 'All A are B' is 'Some A are not B'. It is not 'No A are B', which is a stronger statement.

Question 35Logical Reasoning

The statement 'If it is a cat, then it is a mammal' is true. Which of the following is the converse?

  1. If it is a cat, then it is not a mammal.
  2. If it is a mammal, then it is a cat. Correct answer.
  3. If it is not a cat, then it is not a mammal.
  4. If it is not a mammal, then it is not a cat.

Explanation

The converse of 'If P then Q' is 'If Q then P'. Swapping the hypothesis and conclusion gives the converse.

Question 36Logical Reasoning

Consider the statement 'If a number is divisible by 4, then it is even.' Which of the following is the inverse?

  1. If a number is not divisible by 4, then it is not even. Correct answer.
  2. If a number is not even, then it is not divisible by 4.
  3. If a number is divisible by 4, then it is not even.
  4. If a number is even, then it is divisible by 4.

Explanation

The inverse of 'If P then Q' is 'If not P then not Q'. Here P is 'divisible by 4' and Q is 'even'.

Question 37Logical Reasoning

The statement 'If a shape is a square, then it has four equal sides' is true. Which of the following must also be true?

  1. If a shape does not have four equal sides, then it is not a square. Correct answer.
  2. If a shape is not a square, then it does not have four equal sides.
  3. If a shape has four equal sides, then it is a square.
  4. If a shape is a square, then it does not have four equal sides.

Explanation

The contrapositive of a true conditional is always true. The contrapositive of 'If P then Q' is 'If not Q then not P'.

Question 38Logical Reasoning

In a survey of 50 people, 30 like tea, 25 like coffee, and 10 like both. How many like neither tea nor coffee?

  1. 20
  2. 15
  3. 10
  4. 5 Correct answer.

Explanation

Using the principle of inclusion-exclusion, the number who like at least one is 30 + 25 - 10 = 45. Therefore, the number who like neither is 50 - 45 = 5.

Question 39Logical Reasoning

The statement 'All birds can fly' is false. Which of the following best describes a situation that makes it false?

  1. All birds cannot fly.
  2. No birds can fly.
  3. Some birds can fly.
  4. At least one bird cannot fly. Correct answer.

Explanation

A universal statement 'All A are B' is false if there exists at least one A that is not B. So one bird that cannot fly is enough to make the statement false.

Question 40Logical Reasoning

Two statements are logically equivalent. Which of the following must be true?

  1. They have the same truth value in every row of their truth tables. Correct answer.
  2. They are both false in at least one row.
  3. They have opposite truth values in every row.
  4. They are both true in at least one row.

Explanation

Logical equivalence means that the statements have identical truth values under all possible interpretations, so their truth tables match in every row.

Question 41Logical Reasoning

Given: 'If the team wins, then they celebrate.' and 'If they celebrate, then they rest.' The team does not rest. What logically follows?

  1. The team does not celebrate but wins.
  2. The team does not win. Correct answer.
  3. The team wins.
  4. The team celebrates.

Explanation

From the second conditional and the fact that they do not rest, we can infer they do not celebrate (by contrapositive). Then from the first conditional and the fact that they do not celebrate, we infer they do not win (by contrapositive).

Question 42Logical Reasoning

The statement 'If it is a rose, then it is a flower' is true. Which of the following is logically equivalent to it?

  1. If it is a flower, then it is a rose.
  2. If it is not a rose, then it is not a flower.
  3. If it is not a flower, then it is not a rose. Correct answer.
  4. If it is a rose, then it is not a flower.

Explanation

A conditional statement is logically equivalent to its contrapositive. The contrapositive of 'If P then Q' is 'If not Q then not P'.

Question 43Logical Reasoning

Which of the following is the negation of 'Some students play chess'?

  1. Not all students play chess.
  2. No students play chess. Correct answer.
  3. All students play chess.
  4. Some students do not play chess.

Explanation

The negation of 'Some A are B' is 'No A are B'. It is not 'Some A are not B', because both can be true simultaneously.

Question 44Logical Reasoning

The statement 'If x is even, then x is divisible by 2' is true. Which of the following is the converse?

  1. If x is not even, then x is not divisible by 2.
  2. If x is even, then x is not divisible by 2.
  3. If x is divisible by 2, then x is even. Correct answer.
  4. If x is not divisible by 2, then x is not even.

Explanation

The converse of 'If P then Q' is 'If Q then P'. Here P is 'x is even' and Q is 'x is divisible by 2'.

Question 45Logical Reasoning

Which of the following is the inverse of 'If a person is a teenager, then they are younger than 20'?

  1. If a person is younger than 20, then they are a teenager.
  2. If a person is not younger than 20, then they are not a teenager.
  3. If a person is a teenager, then they are not younger than 20.
  4. If a person is not a teenager, then they are not younger than 20. Correct answer.

Explanation

The inverse of 'If P then Q' is 'If not P then not Q'. Here P is 'a person is a teenager' and Q is 'they are younger than 20'.

Question 46Logical Reasoning

Given that 'P or Q' is false, what can be concluded about P and Q?

  1. Both P and Q are false. Correct answer.
  2. P is true and Q is false.
  3. Both P and Q are true.
  4. P is false and Q is true.

Explanation

An inclusive disjunction 'P or Q' is false only when both P and Q are false. If either were true, the disjunction would be true.

Question 47Logical Reasoning

The statement 'If a figure is a rectangle, then it has four right angles' is true. Which of the following must also be true?

  1. If a figure is a rectangle, then it does not have four right angles.
  2. If a figure has four right angles, then it is a rectangle.
  3. If a figure does not have four right angles, then it is not a rectangle. Correct answer.
  4. If a figure is not a rectangle, then it does not have four right angles.

Explanation

The contrapositive of a true conditional is always true. The contrapositive of 'If P then Q' is 'If not Q then not P'.

Question 48Logical Reasoning

In a class of 40 students, 25 study French, 20 study Spanish, and 12 study both. How many study exactly one of these languages?

  1. 45
  2. 33
  3. 21 Correct answer.
  4. 13

Explanation

The number who study only French is 25 - 12 = 13, and only Spanish is 20 - 12 = 8. The total studying exactly one language is 13 + 8 = 21.

Question 49Logical Reasoning

Consider the statements: 'If it is sunny, then we go to the beach.' and 'If we go to the beach, then we swim.' Which of the following is a valid conclusion if it is sunny?

  1. We do not swim.
  2. It is not sunny.
  3. We swim. Correct answer.
  4. We go to the beach but do not swim.

Explanation

From 'If it is sunny, then we go to the beach' and 'it is sunny', we can infer 'we go to the beach' by modus ponens. Then from 'If we go to the beach, then we swim' and 'we go to the beach', we infer 'we swim'.

Question 50Logical Reasoning

Which of the following is the negation of 'All triangles have three sides'?

  1. No triangles have three sides.
  2. Some triangles have three sides.
  3. All triangles do not have three sides.
  4. Some triangles do not have three sides. Correct answer.

Explanation

The negation of 'All A are B' is 'Some A are not B'. It is not 'No A are B', which is stronger.