Tag: axioms, postulates and theorems

Questions Related to axioms, postulates and theorems

The mathematical statements that are proved are called axioms.

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. True

  2. False

  3. cannot be determined

  4. None of the above

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Correct Option: B
Explanation:

False, statements that are proved are theorems.

A proof is required for :

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition

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Correct Option: C
Explanation:

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. 


Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In a mathematical paper, the term theorem is often reserved for the most important results.
  
So, the correct option is $C$ as a theorem needs a proof.

A lemma is a proven statement used for proving another statement.

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. True

  2. False

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Correct Option: A
Explanation:

'Lemma ' is a subsidiary or intermediate theorem in an argument or proof.

A theorem is:

maths axioms, postulates and theorems conditional statements and converse euclid's postulates introduction to euclid's geometry euclid's fifth postulate

  1. an assumption

  2. always true

  3. always false

  4. sometimes true and sometimes false

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Correct Option: B
Explanation:

A theorem is a Statement (logic) that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements.
A theorem is based on assumption that theorem is true.


Example: In a Euclidean space, the sum of measures of the three angles of any triangles is invariably equal to the straight angle, also as $180^o$

Select the correct statement for the following:

$A$: The angles of an equilateral triangle are equal.
$B$: Angles opposite to two congruent sides of a triangle are congruent.

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. $A$ is a theorem and $B$ is its corollary

  2. $B$ is a theorem and $A$ is its corollary

  3. $A$ and $B$ are both theorems

  4. $A$ and $B$ are both corollaries

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Correct Option: B
Explanation:

$B$ is called the  Triangle theorem.  $A$ can be proved using this theorem, as in an equilateral triangle, all sides are equal and the angles opposite to these congruent sides are Equal. So, $A$ is the corollary of $B$

A result in which the (usually short) proof relies heavily on a given theorem is known as 

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. Conjecture

  2. Corollary

  3. Lemma

  4. Axioms

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Correct Option: B
Explanation:

A result derived from a theorem is its corollary.

A proof is required for a :

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. Postulate

  2. Axiom

  3. Theorem

  4. Definition

Show answer
Correct Option: C
Explanation:

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved. 


Theorem — a mathematical statement that is proved using rigorous mathematical reasoning.  In a mathematical paper, the term theorem is often reserved for the most important results.

So, the correct option is $C$ as a theorem needs a proof.

The formation or expression of an opinion or theory without sufficient evidence for proof is known as

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. axioms

  2. conjecture

  3. corollary

  4. theorem

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Correct Option: B
Explanation:
The formation or expression of an opinion or theory without sufficient evidence for proof is known as conjecture.

For example, make a conjecture about the next number in the pattern $2, 6, 11, 17, ...$. The terms increase by $4$, then $5$, and then $6$.

__________ implies to conclude or suppose from grounds or evidence insufficient to ensure reliability.

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. axioms

  2. conjecture

  3. corollary

  4. theorem

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Correct Option: B
Explanation:
Conjecture implies to conclude or suppose from grounds or evidence insufficient to ensure reliability.

For example, make a conjecture about the next number in the pattern 2 ; 6 ; 11 ; 17 ; ... The terms increase by 4, then 5, and then 6.

What is a conjecture?

maths axioms, postulates and theorems euclid's fifth postulate conditional statements and converse euclid's postulates

  1. A true statement

  2. A statement that you believe is true based upon your observations

  3. A proven statement about something you believe

  4. A statement used by scientists

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Correct Option: B
Explanation:

The formation or expression of an opinion or theory without sufficient evidence for proof is known as conjecture. 


Or a conjecture can be stated as a statement that you believe is true based upon your observations.