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Refer to the relation R on the set {1,2,3,4,5) defined by the rule (x, y) = R if 3 divides x - y
1. List the elements of R
2. List the elements of R-1
3. Find the domain of R
4. Find the range of R
5. Find the domain of R-1
6. Find the range of R-1
Give examples of relations on {1,2,3,4} having the properties specified in the following:
10. Reflexive, antisymmetric, and not transitive
9. Not reflexive, not symmetric, and transitive
8. Reflexive, not symmetric, and not transitive
7. Reflexive, symmetric, and not transitive
Q#1 For any sets π΄, π΅ and πΆ, if π΄ β π΅, then π΄ β© πΆ β π΅ β© πΆ.
Q#2 For any sets π΄, π΅ and πΆ,
(π΄ Γ πΆ) β© (π΅ Γ π·) = (π΄ β© π΅) Γ (πΆ β© π·).
Q#3 Given sets π΄, π΅ and πΆ, prove that
π΄ Γ (π΅ β© πΆ) = (π΄ Γ π΅) β© (π΄ Γ πΆ)
Q#4 Prove that If π΄ and π΅ are sets, then π«(π΄)βπ«(π΅) β π«(π΄βπ΅).
Q#5 Suppose π΄ and π΅ are sets.
If π«(π΄) β π«(π΅),π‘βππ π΄ β π΅.
a. 6n Β³+ 12n Β² + 1, for nΓΒ Β Β Β’ Β Β₯1
b. 3n Β²+ 2n log2 n, for nΓΒ Β Β Β’ Β Β₯1
2. Find the theta notation for the number of times the statement x:=x + 1 is executed.
a. for i:=1 to 2n do x=x+1
b. for i;=1 to 2n do for i;=1 to n do X:=x +1
1. Write the multi-sets of prime factors of given numbers.
a. 150
b. 450
c. 1250
2. Find the cardinalities of each multiset in parts 2-1.
3. Present the application offset and multiset in software engineering? Give specific programming example
Find out reflexive, symmetric and transitive closure of following relation R.Β
R = {(1,2), (2,3), (3,3)}
Question#1
Use algebra of sets to prove the following:
i. (π΅ β π΄) βͺ (πΆ β π΄) = (π΅ βͺ πΆ) β π΄
ii. [(π΅ β π΄) cβ© π΄] β π΄c= π΄
iii. (π΄π βͺ π΅)c β© Ac= β
Question#2
Use Mathematical induction to prove the following generalization of one
of De Morganβs law: βnj=1 π΄j= βnj=1 Aj
Question#3
Prove that (π΄ βͺ π΅ βͺ πΆ) β² = π΄β² β© π΅ β² β© πΆ β²
(a) How many outcomes are possible? List all the possible outcomes.
(b) Consider the following events. List all the outcomes included in each of these events. Mention whether each of these events is a simple or a compound event.
(i) Both students like algebra.
(ii) At most one student likes algebra.
(iii) At least one student likes algebra.
(iv)Neither student likes algebra.
(4) Construct a difference table to predict the next term of each sequence.
(a) 9, 4, 3, 12, 37, 84, ...
(b) 10, 10, 12, 16, 22, 30, ..
Consider the setsAandB,whereA={3,|B|}andB={1,|A|,|B|}.Whatare the sets?
Let A={nβN:20β€n<50} and B={nβN:10<nβ€30}. Suppose C is a set such that CβA and CβB. What is the
largest possible cardinality of C?
#339914 on Dec 2023