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Flashcards: Set Theory

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Determine if the following statement is true or false:

In accordance with primitive concepts and notations in set theory, many axioms are deducible from other axioms.

False

True

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Flashcards: Set Theory

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In accordance with primitive concepts and notations in set theory, many axioms are deducible from other axioms.

Your answer:

True

Correct answer:

False

Explanation:

First recall the primitive concepts and notations for set theory.

"class", "set", "belongs to"

Now, when deciding what constitutes a primitive concept, it is agreed upon in the math world that four main criteria must be met:

1. Undefined terms and axioms should be few.

2. Axioms should NOT be logically deducible from one another unless clearly expressed.

3. Axioms are able to be proved.

4. Axioms must NOT lead to paradoxes.

Thus, the statement in question is false by criteria two.

Flashcards: Set Theory

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What is the correct expression of the relationships between the sets comprised of natural numbers, real numbers, and integers?

$\mathbb{R}\subseteq\mathbb{Z}\subseteq \mathbb{N}$

$\mathbb{N}\subseteq\mathbb{Z}=\mathbb{R}$

$\mathbb{Z}\subseteq\mathbb{N}\subseteq \mathbb{R}$

$\mathbb{N}\subseteq\mathbb{Z}\subseteq \mathbb{R}$

$\mathbb{N}=\mathbb{Z}\subseteq \mathbb{R}$

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Flashcards: Set Theory

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What is the correct expression of the relationships between the sets comprised of natural numbers, real numbers, and integers?

Your answer:

$\mathbb{R}\subseteq\mathbb{Z}\subseteq \mathbb{N}$

Your answer:

$\mathbb{N}\subseteq\mathbb{Z}=\mathbb{R}$

Your answer:

$\mathbb{Z}\subseteq\mathbb{N}\subseteq \mathbb{R}$

Your answer:

$\mathbb{N}=\mathbb{Z}\subseteq \mathbb{R}$

Correct answer:

$\mathbb{N}\subseteq\mathbb{Z}\subseteq \mathbb{R}$

Explanation:

The natural numbers are defined as $\mathbb{N}=\left \{ 0,1,2,3,...\right \}$ , the integers are defined as $\mathbb{Z}=\left \{ ...-3,-2,-1,0,1,2,3,...\right \}$ , and the real numbers ( $\mathbb{R}$ ) are defined as the set of all non-complex numbers. As such, $\mathbb{N}$ is a subset of $\mathbb{Z}$ , and $\mathbb{Z}$ is a subset of $\mathbb{R}$ .

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