Cardinality, ℵ₀, and Cantor’s Diagonal Argument Quiz

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1) The cardinality of a countably infinite set is denoted by:

Explanation

Countably infinite sets all have the same cardinality as the natural numbers. This cardinality is called aleph-null (ℵ₀), the smallest infinite cardinal number.
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About This Quiz
Cardinality, , And Cantors Diagonal Argument Quiz - Quiz

Ready to compare different infinities? This quiz focuses on cardinality symbols like ℵ₀ and 2^ℵ₀, and how they classify sets such as ℕ, ℚ, ℝ, P(ℕ), and various function spaces. You’ll revisit Cantor’s diagonal argument to see why the reals and infinite binary sequences are uncountable, and how assuming a... see moreset is countable can lead to a contradiction. You’ll also work with examples involving functions from ℕ to {0,1}, from {0,1} to ℕ, finite binary strings, and products like A × B when one factor is infinite. By the end, you’ll better understand how different infinite sizes relate, and how diagonal and interleaving arguments are used to prove countability or uncountability. see less

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2) Which of these sets has a cardinality of ℵ₀?

Explanation

The rationals can be arranged in a sequence using a diagonal argument. Since they can be listed, their cardinality is ℵ₀.
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3) If a set A has cardinality ℵ₀, it is:

Explanation

The definition of having cardinality ℵ₀ is precisely that the set is countably infinite.
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4) The cardinality of the real numbers |ℝ| is:

Explanation

Cantor’s diagonalization proves that ℝ cannot be put into one-to-one correspondence with ℕ. Thus, |ℝ| is strictly larger than ℵ₀.
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5) The cardinality of the power set of the natural numbers |P(ℕ)| is equal to:

Explanation

Cantor’s theorem states that the power set of a set has strictly larger cardinality than the set. Thus |P(ℕ)| = 2^ℵ₀.
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6) What is the cardinality of the empty set |∅|?

Explanation

The empty set has no elements. Cardinality measures the number of elements, so its cardinality is exactly 0.
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7) If a set has cardinality ℵ₀, it cannot contain:

Explanation

A countably infinite set cannot contain a strictly larger set. Uncountable subsets would have larger cardinality than the whole set, which is impossible.
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8) Cantor's diagonalization argument is used to prove that a set is:

Explanation

The diagonal argument shows that any attempted enumeration of certain sets (like ℝ or infinite binary sequences) must miss some element, proving uncountability.
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9) To prove the rational numbers ℚ are countable, one can arrange them in a(n):

Explanation

Place numerator–denominator pairs in a grid and list the fractions by moving along diagonals. This ensures every rational number eventually appears.
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10) What is the fundamental assumption in the diagonalization proof?

Explanation

Diagonalization begins by assuming a set can be listed (i.e., assumed countable). The construction then produces an element not in the list, giving a contradiction.
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11) The set of all functions from ℕ to {0,1} is:

Explanation

Each function corresponds to an infinite binary sequence, which Cantor showed to be uncountable.
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12) The set of all functions from {0,1} to ℕ is:

Explanation

A function from {0,1} to ℕ is determined by a pair of natural numbers. Since ℕ × ℕ is countable, this set is also countable.
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13) Which of the following sets is uncountable?

Explanation

Only ℝ among these sets has been proven to be uncountable. ℤ, primes, and ℚ are all countable.
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14) Which of the following sets is countable?

Explanation

There are 2ⁿ binary strings of length n. Taking the union over all n gives a countable union of finite sets, which is countable.
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15) Any infinite set is uncountable.

Explanation

Many infinite sets are countable, such as ℕ and ℚ. Uncountability requires more than mere infinitude.
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16) If A is finite and B is infinite, |A × B| equals:

Explanation

Taking finitely many copies of an infinite set does not change cardinality. Thus |A × B| has the same cardinality as B.
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17) The ability to interleave multiple infinite lists into one is a technique used to prove the:

Explanation

Dovetailing consists of weaving multiple enumerations into a single sequence. This method shows that a countable union of countable sets remains countable.
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The cardinality of a countably infinite set is denoted by:
Which of these sets has a cardinality of ℵ₀?
If a set A has cardinality ℵ₀, it is:
The cardinality of the real numbers |ℝ| is:
The cardinality of the power set of the natural numbers |P(ℕ)| is...
What is the cardinality of the empty set |∅|?
If a set has cardinality ℵ₀, it cannot contain:
Cantor's diagonalization argument is used to prove that a set is:
To prove the rational numbers ℚ are countable, one can arrange them...
What is the fundamental assumption in the diagonalization proof?
The set of all functions from ℕ to {0,1} is:
The set of all functions from {0,1} to ℕ is:
Which of the following sets is uncountable?
Which of the following sets is countable?
Any infinite set is uncountable.
If A is finite and B is infinite, |A × B| equals:
The ability to interleave multiple infinite lists into one is a...
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