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Recent questions in Set Theory & Algebra

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Syllabus: Sets, Relations, Functions, Partial orders, Lattices, Groups.

$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count}&1&1&0&2&0&0&0&0.7&2
\\\hline\textbf{2 Marks Count}&0&1&1&0&1&2&0&0.8&2
\\\hline\textbf{Total Marks}&1&3&2&2&2&4&\bf{1}&\bf{2.3}&\bf{4}\\\hline
\end{array}}}$$

Recent questions in Set Theory & Algebra

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1
Kenneth Rosen Edition 7th Exercise 2.5 Question 40 (Page No. 177)
Show that if $S$ is a set, then there does not exist an onto function $f$ from $S$ to $P(S),$ the power set of $S$. Conclude that $\mid S\mid < \mid P(S)\mid .$ This result is known as Cantor's theorem. [Hint: Suppose such a ... $s$ can exist for which $f (s) = T.]$
Show that if $S$ is a set, then there does not exist an onto function $f$ from $S$ to $P(S),$ the power set of $S$. Conclude that $\mid S\mid < \mid P(S)\mid .$ This result is known as Cantor’s theorem. [Hint: Suppose such a function $f$ existed. Let $T = \{s \in S \mid s \notin f (s)\}$ and show that no element $s$ can exist for which $f (s) = T.]$
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 60 views
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0 answers
2
Kenneth Rosen Edition 7th Exercise 2.5 Question 39 (Page No. 177)
We say that a function is computable if there is a computer program that finds the values of this function. Use question $37$ and $38$ to show that there are functions that are not computable.
We say that a function is computable if there is a computer program that finds the values of this function. Use question $37$ and $38$ to show that there are functions that are not computable.
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 20 views
0 votes
0 answers
3
Kenneth Rosen Edition 7th Exercise 2.5 Question 38 (Page No. 177)
Show that the set of functions from the positive integers to the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between $0$ and $1$ and a subset of ... to the real number $0.\:d_{1}d_{2} \dots d_{n}\dots $ the function $f$ with $f (n) = dn.]$
Show that the set of functions from the positive integers to the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is uncountable. [Hint: First set up a one-to-one correspondence between the set of real numbers between $0$ and $1$ and a subset of these functions. Do this by associating to the real number $0.\:d_{1}d_{2} \dots d_{n}\dots $ the function $f$ with $f (n) = dn.]$
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 19 views
0 votes
0 answers
4
Kenneth Rosen Edition 7th Exercise 2.5 Question 37 (Page No. 177)
Show that the set of all computer programs in a particular programming language is countable. [Hint: A computer program written in a programming language can be thought of as a string of symbols from a finite alphabet.]
Show that the set of all computer programs in a particular programming language is countable. [Hint: A computer program written in a programming language can be thought of as a string of symbols from a finite alphabet.]
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 21 views
0 votes
0 answers
5
Kenneth Rosen Edition 7th Exercise 2.5 Question 36 (Page No. 177)
Show that there is a one-to-one correspondence from the set of subsets of the positive integers to the set real numbers between $0$ and $1$. Use this result and question $34$ and $35$ to conclude that $ℵ_{0} < \mid P(Z^{+})\mid =\mid R\mid.\:[$Hint: Look at the first part of the hint for Exercise $35.]$
Show that there is a one-to-one correspondence from the set of subsets of the positive integers to the set real numbers between $0$ and $1$. Use this result and question $34$ and $35$ to conclude that $ℵ_{0} < \mid P(Z^{+})\mid =\mid R\mid.\:[$Hint: Look at the first part of the hint for Exercise $35.]$
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 18 views
0 votes
0 answers
6
Kenneth Rosen Edition 7th Exercise 2.5 Question 35 (Page No. 177)
Show that there is no one-to-one correspondence from the set of positive integers to the power set of the set of positive integers. [Hint: Assume that there is such a one-to-one correspondence. Represent a subset of the set of ... $ith$ string in the list. Show that this new bit string cannot appear in the list.]
Show that there is no one-to-one correspondence from the set of positive integers to the power set of the set of positive integers. [Hint: Assume that there is such a one-to-one correspondence. Represent a subset of the set of positive integers as an infinite bit string with ... complement of the $ith$ bit of the $ith$ string in the list. Show that this new bit string cannot appear in the list.]
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 8 views
0 votes
0 answers
9
Kenneth Rosen Edition 7th Exercise 2.5 Question 32 (Page No. 177)
Show that when you substitute $(3n + 1)^{2}$ for each occurrence of $n$ and $(3m + 1)^{2}$ for each occurrence of m in the right-hand side of the formula for the function $f (m, n)$ in question $31,$ you ... $Q \times Q \rightarrow Q.$
Show that when you substitute $(3n + 1)^{2}$ for each occurrence of $n$ and $(3m + 1)^{2}$ for each occurrence of m in the right-hand side of the formula for the function $f (m, n)$ in question $31,$ you obtain a one-to-one polynomial function $Z \times Z \rightarrow Z.$ It is an open question whether there is a one-to-one polynomial function $Q \times Q \rightarrow Q.$
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 9 views
0 votes
0 answers
10
Kenneth Rosen Edition 7th Exercise 2.5 Question 31 (Page No. 177)
Show that $Z^{+} \times Z^{+}$ is countable by showing that the polynomial function $f : Z^{+} \times Z^{+}\rightarrow Z^{+}$ with $f(m, n) = \dfrac{(m + n − 2)(m + n − 1)}{2} + m$ is one-to one and onto.
Show that $Z^{+} \times Z^{+}$ is countable by showing that the polynomial function $f : Z^{+} \times Z^{+}\rightarrow Z^{+}$ with $f(m, n) = \dfrac{(m + n − 2)(m + n − 1)}{2} + m$ is one-to one and onto.
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 14 views
0 votes
0 answers
15
Kenneth Rosen Edition 7th Exercise 2.5 Question 26 (Page No. 177)
Use question $25$ to provide a proof different from that in the text that the set of rational numbers is countable. [Hint: Show that you can express a rational number as a string of digits with a slash and possibly a minus sign.]
Use question $25$ to provide a proof different from that in the text that the set of rational numbers is countable. [Hint: Show that you can express a rational number as a string of digits with a slash and possibly a minus sign.]
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 9 views
0 votes
0 answers
16
Kenneth Rosen Edition 7th Exercise 2.5 Question 25 (Page No. 177)
Prove that if it is possible to label each element of an infinite set $S$ with a finite string of keyboard characters, from a finite list characters, where no two elements of $S$ have the same label, then $S$ is a countably infinite set.
Prove that if it is possible to label each element of an infinite set $S$ with a finite string of keyboard characters, from a finite list characters, where no two elements of $S$ have the same label, then $S$ is a countably infinite set.
asked Apr 21 in Set Theory & Algebra Lakshman Patel RJIT 8 views
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