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Discrete Mathematics: Sets, Relations, and Functions, Exercises of Mathematics

Kadi Sarva VishwavidyalayaMathematics

The fundamental concepts of discrete mathematics, including sets, relations, and functions. It provides definitions and examples for various set operations, such as union, intersection, and complement. The document also explores the properties of relations, including reflexivity, symmetry, and transitivity, and introduces the concept of functions, including one-to-one and onto functions. The content is structured into several tasks, each focusing on a specific aspect of discrete mathematics, making it a comprehensive resource for students studying this subject. The document could be useful for university students enrolled in courses related to computer science, mathematics, or other fields that require a strong foundation in discrete mathematics.

Typology: Exercises

2023/2024

Uploaded on 08/27/2024

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Kadi Sarva Vishwavidyalaya
Vidush Somany Institute of Technology and Research, Kadi
B.E. Sem III (CE,CSE, IT)
Sub.: Discrete Mathematics
Unit -1: Set, Relation & Function
Task-1
1) Define 1) set 2) empty set 3) singleton set 4) power set 5) finite set
6) infinite set 7)cardinal number of set .
2) If A = {a, b, c, d}, B = {c, d, e} and C = {e, f, g, h} state the elements of the Sets.
(1) A โˆช C (2) B โˆฉ A (3) B โˆฉ (A โˆช C) (4) (B โˆฉ A) โˆช (B โˆฉ C)
3) For ๐ด={1,2,4,5} & ๐ต={4,5,6,7} & U = {1,2,4,5,6,7} determine the following set.
1. ๐ดโˆ’๐ต
2. ๐ตโˆ’๐ด
3. ๐ด โˆ† ๐ต
4. ๐ดโ€ฒ
5. ๐ตโ€ฒ
4) For universal set U={1,2,3,4,5,6,7,8} & two sets ๐ด ={1,2,3,4} & ๐ต={4,5,6,7,8}
draw Venn diagram for the following condition.
1. ๐ดโ‹ƒ๐ต
pf3
pf4
pf5
pf8

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Download Discrete Mathematics: Sets, Relations, and Functions and more Exercises Mathematics in PDF only on Docsity!

Kadi Sarva Vishwavidyalaya

Vidush Somany Institute of Technology and Research, Kadi

B.E. Sem III (CE,CSE, IT)

Sub.: Discrete Mathematics

Unit - 1: Set, Relation & Function

Task- 1

  1. Define 1) set 2) empty set 3) singleton set 4) power set 5) finite set
  2. infinite set 7)cardinal number of set.
  3. If A = {a, b, c, d}, B = {c, d, e} and C = {e, f, g, h} state the elements of the Sets. (1) A โˆช C (2) B โˆฉ A (3) B โˆฉ (A โˆช C) (4) (B โˆฉ A) โˆช (B โˆฉ C) 3 ) For ๐ด={1,2,4,5} & ๐ต={4,5,6,7} & U = {1,2,4,5,6,7} determine the following set.
  1. ๐ดโˆ’๐ต
  2. ๐ตโˆ’๐ด
  3. ๐ด โˆ† ๐ต
  4. ๐ดโ€ฒ
  5. ๐ตโ€ฒ 4 ) For universal set U={1,2,3,4,5,6,7,8} & two sets ๐ด ={1,2,3,4} & ๐ต={4,5,6,7,8} draw Venn diagram for the following condition.
  6. ๐ดโ‹ƒ๐ต
  1. Define following sets in tabular form. Also show that which of the following are null sets or singleton?
  1. ๐ด={๐‘‹:๐‘‹ ๐œ– โ„ and ๐‘‹ is a solutionof 2 x + 2 = 0 }.
  2. ๐ต={๐‘‹:๐‘‹๐œ– โ„ค and ๐‘‹ ๐‘–๐‘  ๐‘Ž ๐‘ ๐‘œ๐‘™๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐‘œ๐‘“ ๐‘‹โˆ’ 3 = 0 }
  3. ๐ถ={๐‘‹:๐‘‹๐œ– โ„ค and ๐‘‹ ๐‘–๐‘  ๐‘Ž ๐‘ ๐‘œ๐‘™๐‘ข๐‘ก๐‘–๐‘œ๐‘› ๐‘œ๐‘“ x^2 โˆ’ 2 = 0 } 6 ) Let ๐ด={๐‘‹:๐‘‹ is a even natural number less then or equal to 10 } and ๐ต={๐‘‹:๐‘‹ is an odd natural number less then or equal to 10 }. Find (๐‘–)๐ดโˆ’๐ต (๐‘–๐‘–)๐ตโˆ’๐ด (๐‘–๐‘–๐‘–) is ๐ดโˆ’๐ต=๐ตโˆ’๐ด? 7 ) Let โ„• be the universal set and ๐ด, ๐ต, ๐ถ, ๐ท be its subsets given by ๐ด={๐‘‹:๐‘‹ is a even natural number } B = { ๐‘‹:๐‘‹โˆˆ โ„• and X is a multiple of 3 } C = { ๐‘‹:๐‘‹โˆˆ โ„• and X โ‰ฅ 5 } D = { ๐‘‹:๐‘‹โˆˆ โ„• and X โ‰ค 10 }

1. { ๐‘‹: ๐‘‹ ๐œ– โ„ค and x^2 โˆ’ 9 = 0 }

2 .{๐‘ฆ:๐‘ฆ๐œ– โ„• and 1 โ‰ค๐‘ฆ โ‰ค 3 }

  1. Prove that A โˆฉ (B - C) โŠ‚ A โ€“ (B โˆฉ C)
  2. Use the properties of sets to prove that for all the sets A and B , A โ€“ (A โˆฉ B) = A โ€“ B Ans: 2) 12 3) 9 4) 80 , 92 5) 100 Task - 3

1 ) Let A={ 1 , 2 , 3 } & B={a, b} then find ๐ดร— ๐ต, ๐ดร— ๐ด & ๐ตร— ๐ต.

2 ) If A = { 1 , 3 , 5 } and B = { 2 , 3 }, then Find: (i) A ร— B (ii) B ร— A (iii) A ร— A (iv) (B ร— B) 3 ) If A ร— B = {(a, 1 ); (a, 2 ); (b, 1 ); (b, 2 ); (c, 1 ); (c, 2 )}, find A and B. 4 ) If P ร— Q = {(x, 2 ); (x, 6 ); (x, 3 ); (y, 3 ); (y, 6 ); (y, 2 )}, find Q ร— P. 5 ) If A = { 1 , 2 , 3 } and B = { 4 , 5 , 6 }, state which of the following is a relation from A to B.

(a) ๐‘… 1 = { ( 1 , 4 );( 2 , 5 );( 6 , 3 )} (b) ๐‘… 1 = { ( 2 , 5 );( 6 , 3 )}

(c) ๐‘… 1 = { ( 4 , 1 );( 5 , 2 );( 6 , 3 )}

(d) ๐‘… 1 = { ( 1 , 5 );( 1 , 6 );( 2 , 4 );( 2 , 6 );( 3 , 4 );( 3 , 5 )}

  1. Find the domain and range of the relation R where R is a define in (a) and (b)

(a) A = {1, 2, 3, 4, 5} ,B = {1, 2, 3, 10} , aRb if and only if 2a = b. (b) A = {1, 2, 3, 4} = B , aRb if and only if a + b = 5. 7 ) Write the domain and range of the following relations. (a) Rโ‚ = {( 4 , 3 ); ( 6 , 8 ); ( 4 , 8 ); ( 0 , 9 ); ( 7 , 5 ); ( 0 , 10 )} (b) Rโ‚‚ = {(a, 2 ); (b, 3 ); (c, 2 ); (a, 3 ); (d, 4 ); (b, 4 )} Task- 4

  1. For each of the following relation on A = {1, 2, 3, 4},determine whether it is reflexive, symmetric or transitive (1) R = {(1, 4), (4, 1)} (2) R = {(1, 1)} (3) R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3, 2)} (4) R = {(1, 3), (1, 4)}
  2. Let X = {1, 2, 3, 4, 5, 6, 7} and R = {< x, y > /xโˆ’y is divisible by 3} ,then show that R is an equivalence relation.
  3. On the set โ„ค of all integer,define the relation R by R = {(a, b) โˆˆโ„คร—โ„ค/a โˆ’bis divisible by 5 } then show that R is an equivalence relation.
  4. Let A = {1, 2, 3, 4, 5} and let R be a relation on A define

for all x โˆˆโ„š then find the inverse of f if it exist.

  1. Let f : โ„•โ†’โ„• be defined by f(n) = n + 3 for all n โˆˆโ„• then show that f is one-one but not on-to. Task- 6
  2. A = {1, 2, 3, 4} and B = {p, q, r, s} and R = {(1, p), (1, q),(1, r), (2, q), (2, r), (2, s)} then find matrix relation MR.
  3. Let A = {1, 4, 5} and R = {(1, 4), (1, 5), (4, 1), (4, 4), (5, 5)}. Then find matrix relation MR.
  4. Let A= { a,b,c,d } and B = {1,2,3}.Let R be relation define from set A to set B and is given as R = { (a,1),(a,2) ,(b,1),(c,2) ,(d,1)}. Draw a diagraph for R.
  5. Let A={ 1,2,3,4 } and B = { 1,4,6,8,9 }. Let R be a relation from set A to set B and defined as aRb if and only if 2 b = a. Represent this relation in diagraph form.
  6. Let A={ 1,2,3,4,6 } be a set and R be a relation on set A defined as aRb if and only if a is multiple of b. Represent this relation in diagraph form.