Which of the following statements are true? Justify your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so.)
i) If A and B are two sets such that A ⊆ B, then A × B = B.
ii) If S is the set of people on the rolls of IGNOU in 2016 and T is the set of real
numbers lying between 2.5 and 2.55, then SUT is an infinite set.
he set {x∈Z| x ≡1(mod30)}is a group with respect to multiplication(mod30).
iv) If G is a group with an abelian quotient group G/N, then N is abelian.
v) There is a group homomorphism f with Ker f ≅ R and Imf ≅{0}.
Solution:
i) False.
Example: A={1}, B={1,2}
Clearly A ⊆ B, then A × B = {(1,1),(1,2)} B
ii) True.
S is clearly a finite set as they are numbers of rolls in IGNOU.
But T is clearly infinite as there are infinite real numbers between 2.5 and 2.55.
And we know that the union of finite and infinite sets is infinite.
iii) True.
Given set be S
which clearly is a group with respect to multiplication(mod30).
iv) True.
Each element of G/N is a coset aN for some a∈G
Let aN, bN be arbitrary elements of G/N where a,b∈G
Then we have
(aN)(bN)=(ab)N
=(ba)N [Because G is abelian]
=(bN)(aN)
Thus, N is abelian.
v) True.
Example: , where O is zero function, having ker f = R and Im f = {0}
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