NIMCET Set Theory - 28 Essential Flashcards (Full Chapter)

Complete revision deck covering all Set Theory topics asked in NIMCET. Includes: Set definitions, Power Sets, De Morgan's Laws, Inclusion-Exclusion Principle (2 & 3 sets), Cartesian Product, Relations (Reflexive/Symmetric/Transitive/Equivalence), Finite/Infinite sets, and high-yield NIMCET special questions. Perfect for last-minute revision. Master these 28 cards to ace every Set Theory question in the exam!

What is the cardinality of a set? How is it denoted?

The number of distinct elements present in a set. Denoted by n(A) or |A|.

Define the Null/Empty Set. Give one example.

A set containing no elements. Denoted by phi or { }. Example: {x : x in R, x^2 + 1 = 0}.

What is the difference between Equal Sets and Equivalent Sets?

Equal Sets: Exactly the same elements (e.g., {1,2} = {2,1}). Equivalent Sets: Same cardinality (number of elements), but elements can differ (e.g., {1,2} is equivalent to {a,b}).

When is set A called a proper subset of set B?

If every element of A is in B, AND there exists at least one element in B that is not in A. Denoted by A ⊂ B.

What is a Power Set? If |A| = n, what is |P(A)|?

The set of ALL subsets of a given set A. Denoted by P(A). If |A| = n, then |P(A)| = 2^n.

If a set has 4 elements, how many subsets does it have? How many of these are proper subsets?

Total subsets = 2^4 = 16. Proper subsets = 2^4 - 1 = 15 (excluding the set itself).

Which set is the subset of EVERY set?

The Null set (phi).

Define Union, Intersection, and Set Difference mathematically.

A U B = {x : x in A or x in B}. A ∩ B = {x : x in A and x in B}. A - B = {x : x in A and x not in B}.

What is the Symmetric Difference of sets A and B? Write its formula in terms of Union and Intersection.

Elements that belong to exactly one of A or B. Denoted by A △ B. Formula: A △ B = (A - B) U (B - A) = (A U B) - (A ∩ B).

If the universal set is U, what is the complement of set A (A' or A^c)?

All elements of U that are NOT in A. A' = U - A.

State the Idempotent Laws.

1. A U A = A. 2. A ∩ A = A.

State the Identity Laws.

1. A U phi = A. 2. A ∩ U = A (where U is the universal set).

State De Morgan's Laws for sets.

1. (A U B)' = A' ∩ B'. 2. (A ∩ B)' = A' U B'.

State the Distributive Laws.

1. A ∩ (B U C) = (A ∩ B) U (A ∩ C). 2. A U (B ∩ C) = (A U B) ∩ (A U C).

What is the Inclusion-Exclusion Principle for two sets A and B?

n(A U B) = n(A) + n(B) - n(A ∩ B).

What is the Inclusion-Exclusion Principle for THREE sets A, B, and C?

n(A U B U C) = n(A) + n(B) + n(C) - n(A∩B) - n(B∩C) - n(C∩A) + n(A∩B∩C).

If n(A) = 20, n(B) = 30, and n(A U B) = 40, find n(A ∩ B).

Using inclusion-exclusion: n(A ∩ B) = 20 + 30 - 40 = 10.

In a group of 100 people, 60 like tea, 50 like coffee, and 30 like both. How many like exactly one of the two drinks?

Exactly one = n(Only Tea) + n(Only Coffee). Only Tea = 60 - 30 = 30. Only Coffee = 50 - 30 = 20. Total exactly one = 30 + 20 = 50.

Define the Cartesian Product A × B. If |A| = m and |B| = n, what is |A × B|?

A × B = {(a, b) : a in A, b in B}. Cardinality = m × n.

What is a Relation R from set A to set B?

A relation is a subset of the Cartesian product A × B (i.e., R ⊆ A × B).

Define a Reflexive relation. What is the MINIMUM number of elements in a reflexive relation on a set with n elements?

A relation R on set A is reflexive if (a, a) in R for EVERY a in A. Minimum elements = n (the diagonal pairs).

Define Symmetric and Transitive relations.

Symmetric: If (a, b) in R, then (b, a) in R. Transitive: If (a, b) in R and (b, c) in R, then (a, c) in R.

What is an Equivalence Relation?

A relation that is Reflexive, Symmetric, AND Transitive simultaneously.

Is the set of all prime numbers finite or infinite? Is the set of all integers less than -100 finite or infinite?

Primes: Infinite. Integers less than -100: Infinite ({..., -103, -102, -101}).

Let A = {x : x in N, x^2 - 5x + 6 = 0}. Is it finite or infinite? What is its cardinality?

Finite. Solving gives x = 2, 3. So A = {2, 3}. Cardinality = 2.

If A = {4^n - 3n - 1 : n in N} and B = {9(n-1) : n in N}, what is A U B equal to?

4^n - 3n - 1 is always a multiple of 9 (expand via binomial theorem). Therefore A ⊆ B. Hence, A U B = B.

If A ∩ X = B ∩ X = phi and A U X = B U X, then what is the relation between A and B?

Since both A and B are disjoint from X, but give the same union with X, the only possibility is A = B.

What is a set, and what are the two basic criteria for a collection of objects to be called a set?	

A set is a well-defined collection of distinct objects. Criteria: (1) Well-defined (unambiguous criteria for membership) and (2) Distinct (no duplicate elements).