Sets-1-Quiz-9

SETS-1 Quiz-9

Dear Readers,

In mathematics a set is a collection of distinct elements. The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets. Two sets are equal if and only if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century..

Q1. Let A={1,2,3,4}, and let R={(2,2),(3,3),(4,4),(1,2)} be a relation on A. Then, R is

•  Reflexive
•  Symmetric
•  Transitive
•  None of these

Solution
(c) Since (1,1)∉R. So, R is not reflexive Now, (1,2)∈R but, (2,1)∉R. Therefore, R is not symmetric. Clearly, R is transitive

Q2.In a class, 70 students wrote two tests viz; test-I and test-II. 50% of the students failed in test-I and 40% of the students in test-II. How many students passed in both tests?

•  21
•  7 >
•   28
•  14

Solution
(b) Use n(A∪B)=n(A)+n(B)-n(A∩B)

Q3.  Let A_1,A_2,A_3…,A_100 be 100 sets such that n(A_i )=i+1 and A_1⊂A_2⊂A_3⊂⋯⊂A_100, then ⋃_(i=1)^100▒A_i contains… elements

•   99
•  100
•  101
•  102

Solution
(c) We have, A_1⊂A_2⊂A_3⊂⋯⊂A_100 ∴⋃_(i=1)^100▒〖A_i=A_100⇒n(⋃_(i=1)^100▒A_i )=n(A_100 )=101〗

Q4. Let P={(x,y)│x^2+y^2=1,x,y∈R}. Then, P is

•  Reflexive
•  Symmetric
•  Transitive
•  Antisymmetric

Solution
(b) Obviously the relation is not reflexive and transitive but it is symmetric, because x^2+y^2=1⇒y^2+x^2=1

Q5. Set builder form of the relation R={(-2,-7),(-1,-4),(0,-1),(1,2),(2,5)} is

•  R{(a,b):b=2a-3;a,b,∈Z}
•  ((x,y):y=3x-1;x,y∈Z}
•  {(a,b):b=3a-1;a,b∈N}
•  {(u,υ):υ=3u-1;-2≤u }

Solution
 (d) Let R={(x,y):y=ax+b}. Then, (-2,-7),(-1,-4)∈R ⇒-7=-2a+b and -4=-a+b ⇒a=3,b=-1 ∴y=3x-1 Hence, R={(x,y):y=3x-1,-2≤x}

Q6. Let X be the set of all engineering colleges in a state of Indian Republic and R be a relation on X defined as two colleges are related iff they are affiliated to the same university, then R is

•  Only reflexive
•  Only symmetric
• Only transitive
•  Equivalence

Solution
(d) Clearly, R is reflexive symmetric and transitive. So, it is an equivalence relation

Q7. Which of the following cannot be the number of elements in the power set of any finite set?

•  26
•  32
•  8
•  16

Solution
(a) The power set of a set containing n elements has 2^n elements. Clearly, 2^n cannot be equal to 26

Q8. If a N={a x∶x∈N} and b N∩c N=d N, where b,c∈N then

•  d=bc
•  Tc=bd
•  b=cd
•  None of these

Solution
(d) We have, b N={b x│x∈N}= Set of positive integral multiples of b c N={c x│x∈N}= Set of positive integral multiples of c ∴c N={c x | x∈N}= Set of positive integral multiples of b and c both ⇒d=1.c.m.of b and c

Q9. Let A={p,q,r}. Which of the following is not an equivalence relation on A?

•  R_1={(p,q),(q,r),(p,r),(p,p)}
•  R_2={(r,q),(r,p),(r,r),(q,q)}
•  R_3={(p,p),(q,q),(r,r).(p,q)}
•  None of these

Solution
D

Q10. If A is a non-empty set, then which of the following is false? p∶ There is at least one reflexive relation on A q∶ There is at least one symmetric relation on A

•  p alone
•  q alone
•  Both p and q
• Neither p nor q

Solution
(d) The identity relation on a set A is reflexive and symmetric both. So, there is always a reflexive and symmetric relation on a set