Logic and Compound Statements Quiz

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Questions and Answers

Which statement represents a conjunction of two statements?

21 is divisible by 3 and 21 is divisible by 6 (correct)

21 is divisible by 3 if 21 is divisible by 6

21 is not divisible by 6

21 is divisible by 3 or 21 is divisible by 6

How is the statement '21 is not divisible by 3' formally expressed?

not [21 is divisible by 6]

not [21 is divisible by 6] or 21 is divisible by 3

not [21 is divisible by 3 and 21 is divisible by 6]

not [21 is divisible by 3] (correct)

What does the term 'A if and only if B' denote?

B is true only if A is true

Both A and B must be true

A is true only if B is true (correct)

A and B can be true or false independently

When replacing '21' with variable 'x', what changes regarding the statements?

Some statements may become undefined Signup and view all the answers

What is the result of 'not (A or B)'?

not A and not B Signup and view all the answers

What does the logical expression 'if A then B' imply?

B must be true if A is true. Signup and view all the answers

Which of the following statements represents a disjunction?

21 is divisible by 3 or 21 is divisible by 6 Signup and view all the answers

Which statement represents an implication?

21 is divisible by 3 only if 21 is divisible by 6 Signup and view all the answers

Are the statements 'if A then B' and 'if B then A' logically equivalent?

No, they yield different truth values under certain conditions. Signup and view all the answers

What does the phrase 'not A or B' imply when expressed in formal logic?

A may be false or B is true, not necessarily defining both Signup and view all the answers

What is the implication of the truth table where 'if A then B' is false?

A is true and B is false. Signup and view all the answers

What can be said about the statements 'A and B' and 'B and A'?

They are logically equivalent. Signup and view all the answers

Which statement is true regarding the truth values of 'A or B' and 'B or A'?

They are logically equivalent. Signup and view all the answers

Why do some students mistakenly think 'if A then B' is equivalent to 'if B then A'?

They assume logical definitions are order-dependent. Signup and view all the answers

In the context of A iff B, which is true?

Both A and B must be true or both false. Signup and view all the answers

What can be concluded from 'A and B' being the same as 'B and A'?

The order of propositions doesn't affect their truth. Signup and view all the answers

What is the contrapositive of the statement 'If A then B'?

If not B then not A Signup and view all the answers

Which pair of statements is logically equivalent?

If two triangles have the same angles then they are similar, if two triangles are not similar then they do not have the same angles Signup and view all the answers

Why is it incorrect to write the contrapositive of 'If a and b are odd, then ab is odd' as 'If ab is not odd then a and b are not odd'?

It misrepresents the relationship between the conditions. Signup and view all the answers

What is the contrapositive of the statement 'If 𝑥 > 4 then 𝑥^2 > 16'?

If 𝑥^2 ≤ 16 then 𝑥 ≤ 4 Signup and view all the answers

Which statement indicates a misconception about contrapositives?

The contrapositive of a true statement is always false. Signup and view all the answers

What is the converse of the statement 'If A then B'?

If B then A Signup and view all the answers

What does 'not B only if not A' state?

If B is false, then A must also be false. Signup and view all the answers

Which of the following statements correctly demonstrates a contrapositive?

If a student studies then they will pass is equivalent to If a student does not pass then they did not study. Signup and view all the answers

Under what conditions is the statement A iff B considered true?

When both A and B are true or both are false Signup and view all the answers

What does the phrase 'A if and only if B' imply?

If A is true, then B must also be true, and if B is true, then A must also be true Signup and view all the answers

Which logical equivalence is correctly associated with 'A if B'?

If B then A Signup and view all the answers

In constructing a truth table for A iff B, which of the following pairs of truth values yield a true result?

T, T and F, F Signup and view all the answers

What representations can be used to visualize A iff B?

Diagrams for A if B and B if A combined with 'and' Signup and view all the answers

The statement 'An integer is divisible by 9 if and only if the sum of its digits is divisible by 9' is an example of:

A biconditional statement Signup and view all the answers

When constructing a truth table, what does the row marked F, F represent in the context of A iff B?

Both A and B are false, making A iff B true Signup and view all the answers

How are 'A only if B' and 'A if B' logically distinguished?

A only if B checks the necessity of A but does not reciprocate Signup and view all the answers

Which symbol represents 'for all' in mathematical notation?

∀ Signup and view all the answers

What does the notation ∃𝑥 ∈ ℝ: 𝑥 > 0 indicate?

There exists a real number such that it is greater than zero. Signup and view all the answers

Which of the following is a correct interpretation of the notation ∀𝑥 ∈ ℝ, 𝑥 ≤ 0?

All real numbers are less than or equal to zero. Signup and view all the answers

What is the difference between ∀ … ∃ and ∃ … ∀?

The order of quantifiers affects the truth of the statement. Signup and view all the answers

In mathematical proof, what is essential for a proof to be considered rigorous?

Obeying mathematical and logical rules consistently. Signup and view all the answers

Which statement correctly reflects the essence of a mathematical proof?

It can range from a single line to very complex explanations adhering to rules. Signup and view all the answers

Which symbol represents ‘there exists’ in mathematical notation?

∃ Signup and view all the answers

If a statement includes the phrase 'it is not the case that for all real x, x > 0', how can it be represented in symbols?

∃𝑥 ∈ ℝ: 𝑥 ≤ 0 Signup and view all the answers

Study Notes

Compound Statements

Compound statements are constructed by combining other statements using logical connectives.
and, or, not, if...then, if and only if are logical connectives.
The truth value of a compound statement depends on the truth or falsity of the statements combined.

Negation

not is a logical connective that negates a statement.
If statement A is true, then not A is false, and vice versa.
not applies only to the statement immediately after it, unless there are brackets.

If and Only If

A if and only if B (A iff B) is a compound statement equivalent to (A if B) and (A only if B).
A if B is logically equivalent to if B then A.
A only if B is logically equivalent to if B then A.
A iff B is true when A and B are both true, or both false.
A iff B means A and B always say the same thing.

Swapping A and B

A and B is logically equivalent to B and A.
A or B is logically equivalent to B or A.
if A then B is not logically equivalent to if B then A.

Contrapositive

The contrapositive of if A then B is if not B then not A.
The contrapositive is logically equivalent to the original statement.

Universal Quantifier

for all is written as ∀.
∀x ∈ ℝ represents "for all real numbers x."

Existential Quantifier

there exists is written as ∃.
∃x ∈ ℝ represents "there exists a real number x."

Mathematical Proof

A proof is an explanation of why a statement is true.
A proof must be rigorous and convincing.
Proofs can vary in length depending on the complexity of the statement.

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Description

Test your understanding of compound statements and logical connectives. This quiz covers key concepts such as negation, 'if and only if', and the relationships between different logical statements. Challenge yourself to apply these concepts to various scenarios.