c. 2.4 Undefined Notions and Axioms of Set Theory 45 Undefined Notions and Axiom

Question

c. 2.4 Undefined Notions and Axioms of Set Theory 45 Undefined Notions and Axioms of Set Theory 2.4 Questions to guide your reading of this section: 1. What are undefined notions and arioms? 2. What role do axioms serve relative to umndefined notions? 3. What are the two undefined notions of set theory? 4. What axioms are we assuming in our study of sets? What does each cf these axioms tell us? An approach to mathematics that relies exclusively on pictures, intuition, and lexperience ultimately puts at risk the goals of objective certainty and precise communication. At some point, we want to be certain that the conclu- we have reached in our investigations are correct. And we want to be able t indinm and our reasoning in a precise

Explanation / Answer

Question 1.

Undefined Notions:

At the beggining of any formal mathematical study , we generally do not have notations, terminology and official record but we have ideas only which are worthful to the corresponding study. We collect certain experience of working with these ideas and some intuitive knowledge for these ideas on how they might relate with another . These ideas carry no intrinsic meaning although the words or symbols used to represent them often convey some intuitive concept. These very beggining ideas that are formally introduced , without definitions , are known as undefined notions.

Axioms :

In short words , the Axioms are the mathematical assumptions describing the Undefined notions into a relation and having no mathematical proof .

Question 2.

Role of Axioms relative to Undefined notions:

Axiom describes the Undefined Notions and contribute on how these undefined notions relate to each other.

Question 3.

Two undefined notions in set theory:

Two Undefined Notions in Set Theory are   - " Set "    and " belongs to "

Question 4.

Axioms in set theory :

1.

Given any set A and any object x , either x belongs to A or x does not belongs to A.

2.

Given any sets , there is a set, called the union of the given sets , whose members are precisely those objects that are members of at least one of the given sets.

3.

Given any object x, we can form a set {x} having x as its only memeber .

4.

Given a set A and a sentence P(x) that is unambiguously true or unambiguously false depending on the particular member x of A , there exists a set {x belongs to A : P(x) } whose members are precisely those members x of A for which P(x) is true.

5.

There is a set having no member.

Explanation :

Axiom 1 . talks of fundamental assumption of set membership .

Axiom 2. talks about formation of Union of sets.

Axiom 3. talks about formation signleton set .

Axiom 4. talks about formation of sets using set-builder notation .

Axiom 5. talks about existence of Empty set .

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