Day 01- Discrete Mathematics for gate CSEIT- Concept of empty set or null set.<\/p>\n
[fvplayer id=”276″]<\/p>\n
An empty set<\/strong> (also called a null set<\/strong>) is a set that contains no elements<\/strong>. It is denoted by: \u2714 Subset Property:<\/strong> The empty set is a subset of every set<\/strong>. Q1:<\/strong> What is the power set of the empty set? Answer:<\/strong> \u00a0(A) {\u2205}<\/strong><\/p>\n Trick:<\/strong> The power set contains all subsets of a set. Since \u2205 has no elements, its only subset is itself!<\/p>\n The empty set (\u2205)<\/strong> is a fundamental concept in Discrete Mathematics<\/strong> and is used in set theory, logic, and algorithms<\/strong>. It plays a key role in solving problems related to relations, functions, and probability<\/strong> in GATE CSE\/IT<\/strong>.<\/p>\n Great! Let\u2019s break down the concept of the empty set (null set)<\/strong> from Day 01 of Discrete Mathematics for GATE CSE\/IT<\/strong> in a simple, beginner-friendly way.<\/p>\n An empty set<\/strong> (also called null set<\/strong>) is a set that has no elements<\/strong>.<\/p>\n \u201cA set which does not contain any element is called an empty set or null set.\u201d<\/p>\n<\/blockquote>\n It is denoted by: Set of prime numbers less than 2 \u2192 \u2205\\emptyset<\/span>\u2205<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n Set of natural numbers less than 0 \u2192 \u2205\\emptyset<\/span>\u2205<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n Set of students in a class who scored more than 100 out of 100 \u2192 \u2205\\emptyset<\/span>\u2205<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ol>\n Cardinality = number of elements = 0 The empty set is a subset of every set<\/strong> Power set = Set of all subsets<\/p>\n<\/li>\n Power set of \u2205\\emptyset<\/span>\u2205<\/span><\/span><\/span><\/span>: Q. What is the power set of the empty set?<\/strong> Q. Is empty set a subset of itself?<\/strong> Q. What is the cardinality of P(P(\u2205))P(P(\\emptyset))<\/span>P<\/span>(<\/span>P<\/span>(<\/span>\u2205<\/span>))<\/span><\/span><\/span><\/span>?<\/strong><\/p>\n First, P(\u2205)={\u2205}P(\\emptyset) = \\{ \\emptyset \\}<\/span>P<\/span>(<\/span>\u2205<\/span>)<\/span>=<\/span><\/span>{<\/span>\u2205<\/span>}<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n Then, P(P(\u2205))={\u2205,{\u2205}}P(P(\\emptyset)) = \\{ \\emptyset, \\{ \\emptyset \\} \\}<\/span>P<\/span>(<\/span>P<\/span>(<\/span>\u2205<\/span>))<\/span>=<\/span><\/span>{<\/span>\u2205<\/span>,<\/span>{<\/span>\u2205<\/span>}}<\/span><\/span><\/span><\/span> \u275d Think of the empty set as an empty box<\/strong> \u2014 it\u2019s still a box (a valid set), but it holds nothing inside. \u275e<\/p>\n<\/blockquote>\n
\u2714 \u2205<\/strong> (phi symbol)
\u2714 {}<\/strong> (curly brackets with nothing inside)<\/p>\n\u00a0Example:<\/strong><\/h4>\n
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\u00a0Important Properties of Empty Set<\/strong><\/h3>\n
\u2714 Cardinality:<\/strong> The number of elements in an empty set is 0<\/strong> \u2192 |\u2205| = 0<\/strong>.
\u2714 Union Property:<\/strong> Any set A<\/strong> united with an empty set remains unchanged \u2192 A \u222a \u2205 = A<\/strong>.
\u2714 Intersection Property:<\/strong> Any set A<\/strong> intersected with an empty set gives an empty set \u2192 A \u2229 \u2205 = \u2205<\/strong>.
\u2714 Power Set:<\/strong> The power set of an empty set is P(\u2205) = {\u2205}<\/strong> (a set containing only the empty set).<\/p>\n\u00a0GATE Practice Question<\/strong><\/h3>\n
(A) {\u2205}
(B) \u2205
(C) {\u2205, {\u2205}}
(D) {{\u2205}}<\/p>\n\u00a0Summary:<\/strong><\/h3>\n
Day 01- Discrete Mathematics for gate CSEIT- Concept of empty set or null set.<\/a><\/h3>\n
Discrete Mathematics for Computer Science<\/a><\/h3>\n
UNIT \u2013 I \u2013 Set Theory \u2013 SMT5201<\/a><\/h3>\n
Empty set<\/a><\/h3>\n
\n\ud83d\udcd8 Day 01 \u2013 Discrete Mathematics for GATE CSE\/IT<\/strong><\/h1>\n
Topic: Concept of Empty Set (Null Set)<\/strong><\/h2>\n
\n\ud83d\udd39 What is an Empty Set?<\/strong><\/h2>\n
\ud83d\udd0d Definition:<\/h3>\n
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\u2705 Symbol:<\/h3>\n
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\u2205or{}\\emptyset \\quad \\text{or} \\quad \\{\\} <\/span>\u2205<\/span>or<\/span><\/span>{<\/span>}<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n
\n\ud83d\udd22 Examples of Empty Sets<\/strong><\/h2>\n
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\n\u26a0\ufe0f Important Notes for GATE:<\/strong><\/h2>\n
1. Cardinality of Empty Set:<\/strong><\/h3>\n
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So, \u2223\u2205\u2223=0|\\emptyset| = 0<\/span>\u2223\u2205\u2223<\/span>=<\/span><\/span>0<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n2. Subset Property:<\/strong><\/h3>\n
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\u2205\u2286Afor\u00a0any\u00a0set\u00a0A\\emptyset \\subseteq A \\quad \\text{for any set } A<\/span>\u2205<\/span>\u2286<\/span><\/span>A<\/span>for\u00a0any\u00a0set\u00a0<\/span><\/span>A<\/span><\/span><\/span><\/span><\/p>\n<\/li>\n<\/ul>\n3. Power Set of Empty Set:<\/strong><\/h3>\n
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P(\u2205)={\u2205}P(\\emptyset) = \\{ \\emptyset \\}<\/span>P<\/span>(<\/span>\u2205<\/span>)<\/span>=<\/span><\/span>{<\/span>\u2205<\/span>}<\/span><\/span><\/span><\/span>
So, cardinality of power set = 1<\/p>\n<\/li>\n<\/ul>\n
\n\ud83d\udcd8 Common GATE MCQ Concept:<\/h2>\n
\ud83d\udc49 Answer: {\u2205}\\{ \\emptyset \\}<\/span>{<\/span>\u2205<\/span>}<\/span><\/span><\/span><\/span><\/p>\n
\ud83d\udc49 Answer: Yes<\/p>\n\n
\ud83d\udc49 So, cardinality = 2<\/p>\n<\/li>\n<\/ul>\n
\n\ud83e\udde0 Easy Trick to Remember:<\/h2>\n
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\n\ud83d\udcdd Summary:<\/h2>\n
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\n \nConcept<\/th>\n Value<\/th>\n<\/tr>\n<\/thead>\n \n Empty Set Notation<\/td>\n \u2205\\emptyset<\/span>\u2205<\/span><\/span><\/span><\/span> or {}\\{\\}<\/span>{<\/span>}<\/span><\/span><\/span><\/span><\/td>\n<\/tr>\n \n Cardinality<\/td>\n 0<\/td>\n<\/tr>\n \n Is subset of all sets?<\/td>\n Yes<\/td>\n<\/tr>\n \n Power Set<\/td>\n {\u2205}\\{ \\emptyset \\}<\/span>