Mathematical Methods for Economics-I

Questions and Answers

Which professor is affiliated with Jawaharlal Nehru University?

Prof.M.S.Bhatt

Prof.Indrani Roy Choudhary (correct)

Prof.B.S.Prakash

Prof.Anup Chatterjee

Which of the following professors is a retired associate professor?

Prof.G.Pradhan

Prof.Kastuva Barik

Dr.Surajit Das

Prof.Anup Chatterjee (correct)

What is the main affiliation of Shri B.S.?

Researcher at JNU

Reputed Economist in New Delhi

Not specified (correct)

Assistant Professor at Mata Sundri College

Which university is associated with Dr.Manjula Singh?

University of Delhi Signup and view all the answers

Which of the following professors is not stated as retired?

Prof.B.S.Prakash Signup and view all the answers

What is defined as a set of ordered pairs?

A relation Signup and view all the answers

Which property is NOT required for a relation to be an equivalence relation?

Injective Signup and view all the answers

What describes a function where every element of the domain has a unique image in the codomain?

Injective function Signup and view all the answers

In the context of functions, what does surjective mean?

Every element in the codomain is covered Signup and view all the answers

What is the set of all first elements in a relation called?

Domain Signup and view all the answers

What does the notation in set-builder form, such as {x | x is a product manufactured by the company R}, indicate?

It signifies all elements x that meet the specified condition. Signup and view all the answers

Which of the following sets is represented by Q+?

The set of all positive rational numbers. Signup and view all the answers

What is a statement in logic?

A declarative sentence that can be either true or false. Signup and view all the answers

What symbol is used to denote an empty set?

ɸ Signup and view all the answers

Which of the following is an example of a command?

Close the window. Signup and view all the answers

What is the definition of a relation in the context of sets?

Any subset of a Cartesian product of sets. Signup and view all the answers

Which of the following sets represents all integers, including both positive and negative values?

Z Signup and view all the answers

Which connective represents 'and' in logical statements?

Conjunction Signup and view all the answers

What does the converse of a statement refer to?

Flipping the hypothesis and conclusion. Signup and view all the answers

In the notation R = { x ∈ R : x > 0 }, what does R represent?

A set of all positive real numbers. Signup and view all the answers

What do necessary and sufficient conditions relate to in logic?

The implication relationship in arguments. Signup and view all the answers

What characteristic defines a generic element in a set when using set-builder notation?

It must follow a specific property or condition. Signup and view all the answers

Which quantifier expresses the idea of 'for all'?

Universal quantifier Signup and view all the answers

Which of these sets is correctly defined as the set of all rational numbers?

Q = { p/q : p, q ∈ Z } Signup and view all the answers

What is the primary purpose of a proof in logic?

To establish the validity of a claim from axioms and assumptions. Signup and view all the answers

What type of proof assumes the opposite of what is to be proven?

Proof by contradiction Signup and view all the answers

What is the meaning of the expression ‘∃ u ∈ R:∀s ∈ S|s ≤ u’?

There exists an upper limit u for all elements in S. Signup and view all the answers

What does the statement ‘for all x ∈Z, there exists y ∈ Z such that y > x’ imply?

All integers have a greater integer. Signup and view all the answers

Which of the following statements about quantifiers is true?

Existential quantifiers assert the existence of at least one example. Signup and view all the answers

In the context of logic, what does an implication represent?

A cause-and-effect relationship. Signup and view all the answers

What is necessary for a set S to have an upper bound?

There must exist a real number u that exceeds every element in S. Signup and view all the answers

How would you define the inverse of an implication?

The negation of both the hypothesis and the conclusion. Signup and view all the answers

What does it mean if a statement is classified as 'sufficient'?

It guarantees the conclusion under certain circumstances. Signup and view all the answers

Which term is used to refer to claims and propositions in mathematics?

Assertions Signup and view all the answers

What is a common role of quantifiers in logic?

To generalize or specify conditions of statements. Signup and view all the answers

Which statement correctly represents a necessary condition?

B must be true for A to be true. Signup and view all the answers

Study Notes

Set Notation and Set Builder Method

Sets represent collections of distinct objects, denoted with curly braces, e.g., P = {x | x is a product manufactured by company R}.
The set-builder notation employs a colon (:) or vertical bar (|) to specify the properties of elements in a set.
Standard notations for commonly used sets:
- N: set of all natural numbers {1, 2, 3, ...}
- Z: set of all integers {0, ±1, ±2, ...}
- Q: set of all rational numbers {p/q | p, q ∈ Z}
- R: set of all real numbers
- C: set of all complex numbers {x | x = a + ib, a, b ∈ R}
An empty set, denoted by the symbol ɸ, contains no elements.

Relations and Functions

A relation is defined as any subset of a Cartesian product of sets, represented as a set of ordered pairs.
For sets X and Y, relations can be expressed as ρ ⊆ X × Y.
A function from X to Y assigns each element in X to a unique element in Y, fulfilling f: X → Y.
Surjective (onto) functions have their range equal to the codomain, meaning every element in Y is covered by the mapping.

Logic and Statements

A statement is a declarative sentence that can be classified as either true or false.
The concept of negation involves rephrasing a statement to reflect its opposite truth value.
Truth tables are used to assess the validity of statements involving logical connectives (e.g., "and", "or").

Quantifiers

Quantifiers express the extent of the variable in a statement:
- Existential quantifier (∃): Indicates the existence of at least one element in a set satisfying a condition.
- Universal quantifier (∀): States that a condition holds for all elements in a set.

Implications and Conditions

Implications relate conditions to conclusions, with terms like necessary and sufficient conditions.
The inverse of a statement involves flipping its premise and conclusion, while the converse flips their order.
"If and only if" statements establish equivalence between two conditions.

Theorems and Proofs

Theorem: A statement that has been proven based on previously established statements (axioms/lemmas).
Lemma: A preliminary proposition used to help prove a theorem.
Proofs can be constructed in various forms, including direct proof, proof by contradiction, proof by induction, and proof using the contrapositive.

Importance in Economics

Understanding these mathematical foundations is crucial for analyzing economic theories and models.
The ability to systematically construct arguments, and differentiate among various mathematical entities is essential for sound economic reasoning.

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This quiz covers key concepts and methods in Mathematical Methods for Economics-I, tailored for students of the School of Social Sciences at Indira Gandhi National Open University. Explore various mathematical tools and their applications in economic theory and analysis.