Algebraic Structures

Let $R$ be a binary relation on the set $\{1,2, \ldots, 10\}$, where $(x, y) \in, R$ if the product of $x$ and $y$ is square of an integer. Which of the following properties is/are satisfied by $R$ ?

$g(.)$ is a function from A to B, $f(.)$ is a function from B to C, and their composition defined as $f(g(.))$ is a mapping from A to C. If $f(.)$ and $f(g(.))$ are onto (surjective) functions, which ONE of the following...

Let $F$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\preccurlyeq$ on $F$ as follows: $\forall f . g \in F, f \preccurlyeq g$ if and only if $\forall x \in\{1, \ldots, n\}...

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \mathrm{~F}$ we define $$\left(f_1 \odot f_2\right)(n)=f_1(n)+f_2(n)$$...

For positive non-zero real variables $p$ and $q$, if $\log \left(p^2 + q^2\right) = \log p + \log q + 2 \log 3$, then, the value of $\frac{p^4 + q^4}{p^2 q^2}$ is

Consider the operators $\diamond$ and $\square$ defined by $a \diamond b=a+2 b, a \square b=a b$, for positive integers. Which of the following statements is/are TRUE?

Let $P$ be the partial order defined on the set {1,2,3,4} as follows: $P = \{(x, x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on {1,2,3,4} that contain $P$ is _________.

Let Z n be the group of integers {0, 1, 2, ..., n − 1} with addition modulo n as the group operation. The number of elements in the group Z 2 × Z 3 × Z 4 that are their own inverses is __________.

Let X be a set and 2$$^X$$ denote the powerset of X. Define a binary operation $$\Delta$$ on 2$$^X$$ as follows: $$A\Delta B=(A-B)\cup(B-A)$$. Let $$H=(2^X,\Delta)$$. Which of the following statements about H is/are corr...

$$f(x)$$ and $$g(y)$$ are functions of x and y, respectively, and $$f(x)=g(y)$$ for all real values of x and y. Which one of the following options is necessarily TRUE for al x and y?

Let $$f:A \to B$$ be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation $$\sim$$ on the set A as $${a_1} \sim {a_2}$$ if $$f({a_1}) = f({a_2})$$, where $${a_1},{a_2} \in A$$...

Which of the following statements is/are TRUE for a group G?

Let r be a root of the equation x 2 + 2x + 6 = 0. Then the value of the expression (r + 2) (r + 3) (r + 4) (r + 5) is

A relation R is said to be circular if a R b and b R c together imply c R a. Which of the following options is/are correct?

Consider the following sets, where n > 2: S 1 : Set of all n x n matrices with entries from the set {a, b, c} S 2 : Set of all functions from the set {0,1, 2, ..., n 2 — 1} to the set {0, 1, 2} Which of the following cho...

Six students P, Q, R, S, T and U, with distinct heights, compare their heights and make the following observations: Observation I : S is taller than R. Observation II : Q is the shortest of all. Observation III : U is ta...

Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.

Let $$G$$ be a finite group on $$84$$ elements. The size of a largest possible proper subgroup of $$G$$ is ________.

A binary relation $$R$$ on $$N \times N$$ is defined as follows: $$(a,b)R(c,d)$$ if $$a \le c$$ or $$b \le d.$$ Consider the following propositions: $$P:$$ $$R$$ is reflexive $$Q:$$ $$R$$ is transitive Which one of the f...

Let $$𝑅$$ be the relation on the set of positive integers such that $$aRb$$ if and only if $$𝑎 $$ and $$𝑏$$ are distinct and have a common divisor other than $$1.$$ Which one of the following statements about $$𝑅$$ i...

The number of onto functions (subjective functions) from set $$X = \left\{ {1,2,3,4} \right\}$$ to set $$Y = \left\{ {a,b,c} \right\}$$ is __________________.

Let $$R$$ be a relation on the set of ordered pairs of positive integers such that $$\left( {\left( {p,q} \right),\left( {r,s} \right)} \right) \in R$$ if and only if $$p - s = q - r.$$ Which one of the following is true...

If $$g(x)=1-x$$ & $$h\left( x \right) = {x \over {x - 1}}\,\,$$ then $$\,\,{{g\left( {h\left( x \right)} \right)} \over {h\left( {g\left( x \right)} \right)}}\,\,\,$$ is

let $$G$$ be a group with $$15$$ elements. Let $$L$$ be a subgroup of $$G$$. It is known that $$L \ne G$$ and that the size of $$L$$ is at least $$4$$. The size of $$L$$ is ______.

Consider the set of all functions $$f:\left\{ {0,\,1,.....,2014} \right\} \to \left\{ {0,\,1,.....,2014} \right\}$$ such that $$f\left( {f\left( i \right)} \right) = i,\,\,\,$$ for all $$0 \le i \le 2014.$$ Consider the...

There are two elements $$x, y$$ in a group $$\left( {G,\, * } \right)$$ such that every elements in the group can be written as a product of some number of $$x's$$ and $$y's$$ in some order. It is known that $$x * x = y...

Let S denote the set of all functions $$f:\,{\{ 0,\,1\} ^4}\, \to \,\{ 0,\,1\} $$. Denote by N the number of functions from S to the set {0, 1}. The value of $${\log _2}$$ $${\log _2}$$ N is___________________

A Binary operation $$ \oplus $$ on a set of integers is defined as $$x$$ $$ \oplus $$ $$y$$ $$ = {x^2} + {y^2}$$. Which one of the following statements is TRUE about $$ \oplus $$ ?

Consider the set $$S = \left\{ {1,\,\omega ,\,{\omega ^2}} \right\},$$ where $$\omega $$ and $${{\omega ^2}}$$, are cube roots of unity. If $$ * $$ denotes the multiplication operation, the structure $$\left\{ {S,\, * }...

consider the binary relation $$R = \left\{ {\left( {x,y} \right),\,\left( {x,z} \right),\,\left( {z,x} \right),\,\left( {z,y} \right)} \right\}$$ on the set $$\left\{ {x,\,y,\,z} \right\}$$. which one of the following is...

Which one of the following in NOT necessarily a property of Group?

How many different non-isomorphic Abelian groups of order 4 are there?

What is the maximum number of different Boolean functions involving $$n$$ Boolean variables?

Let $$S$$ be a set6 of $$n$$ elements. The number of ordered pairs in the largest and the smallest equivalence relations on $$S$$ are

A partial order P is defined on the set of natural numbers as following. Herw x/y denotes integer division. i) (0, 0) $$ \in \,P$$. ii) (a, b) $$ \in \,P$$ if and only a % $$10\, \le $$ b % 10 and )a/10, b/10) $$ \in \,P...

A relation $$R$$ is defined on ordered pairs of integers as follows: $$\left( {x,y} \right)R\left( {u,v} \right)\,if\,x < u$$ and $$y > v$$. Then $$R$$ is

Consider the set S = {a, b, c, d}. Consider the following 4 partitions $$\,{\pi _1},\,{\pi _2},\,{\pi _3},\,{\pi _4}$$ on $$S:\,{\pi _1} = \left\{ {\overline {a\,b\,c\,d} } \right\},\,{\pi _2} = \left\{ {\overline {a\,b\...

For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an identity for $$f$$ if $$f\left( {a,z} \right) = a = f\left( {z,a} \right)$$ for all $$a \in N$$. Whi...

The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Given below are four plausible reasons. Which one of them is false?

The set $$\left\{ {1,\,\,2,\,\,4,\,\,7,\,\,8,\,\,11,\,\,13,\,\,14} \right\}$$ is a group under multiplication modulo $$15$$. The inverse of $$4$$ and $$7$$ are respectively:

Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$$\left[ {\matrix{ a & f & e \cr 0 & b & d \cr 0 & 0 & c \cr } } \right]$$$ Where $$a, b, c, d, e$$ and $$f$$ are real numbers and $$abc$$ $$ \ne \,\,...

Let $$f$$ be a function from a set $$A$$ to a set $$B$$, $$g$$ a function from $$B$$ to $$C$$, and $$h$$ a function from $$A$$ to $$C$$, such that $$h\left( a \right) = g\left( {f\left( a \right)} \right)$$ for all $$a \...

Let R and S be any two equivalence relations on a non-emply set A. Which one of the following statements is TRUE?

Let f: $$\,B \to \,C$$ and g: $$\,A \to \,B$$ be two functions and let h = fog. Given that h is an onto function which one of the following is TRUE?

The inclusion of which of the following sets into S = { {1, 2}, 1, 2, 3}, {1, 3, 5}, {1, 2, 4}, {1, 2, 3, 4, 5} } Is necessary and sufficient to make S a complete lattice under the partial order defined by set containmen...

Consider the binary relation: $$S = \left\{ {\left( {x,y} \right)|y = x + 1\,\,and\,\,x,y \in \left\{ {0,1,2,...} \right\}} \right\}$$ The reflexive transitive closure of $$S$$ is

(a) $$S = \left\{ { < 1,2 > ,\, < 2,1 > } \right\}$$ is binary relation on set $$A = \left\{ {1,2,3} \right\}$$. Is it irreflexive? Add the minimumnumber of ordered pairs to $$S$$ to make it an $$\,\,\,\,\,$$equivalence...

The binary relation $$S = \phi $$ (emply set) on set A = {1, 2, 3} is

Consider the following relations: $${R_1}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is even over the set of integers $${R_2}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$ is odd...

A relation R is defined on the set of integers as zRy if f (x + y) is even. Which of the following statements is true?

Let $$S = \left\{ {0,1,2,3,4,5,6,7} \right\}$$ and $$ \otimes $$ denote multiplication modulo $$8$$, that is, $$x \otimes y = \left( {xy} \right)$$ mod $$8$$ (a) Prove that $$\left( {0,\,1,\, \otimes } \right)$$ is not a...

(a) Mr. X claims the following: If a relation R is both symmetric and transitive, then R is reflexive. For this, Mr. X offers the following proof. "From xRy, using symmetry we get yRx. Now because R is transitive, xRy an...

Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q = q * p,$$, and (b) $$q * q = q$$

The number of functions from an $$m$$ element set to an $$n$$ element set is

Suppose $$A$$ is a finite set with $$n$$ elements. The number of elements in the Largest equivalence relation of $$A$$ is

Let (A, *) be a semigroup. Furthermore, for every a and b in A, if $$a\, \ne \,b$$, then $$a\,*\,b \ne \,\,b\,*\,a$$. (a) Show that for every a in A a * a = a (b) Show that for every a, b in A a * b * a = a (c) Show that...

Suppose A = {a, b, c, d} and $${\Pi _1}$$ is the following partition of A $${\Pi _1}\, = \,\{ \{ a,\,\,b,\,\,c\,\} \,,\,\{ d\} \,\} $$ (a) List the ordered pairs of the equivalence relations induced by $${\Pi _1}$$ (b) D...

Let $${R_1}$$ and $${R_2}$$ be two equivalence relations on a set. Consider the following assertions: (i)$$\,\,\,\,{R_1} \cup {R_2}$$ is an euivalence relation (ii)$$\,\,\,\,{R_1} \cap {R_2}$$ is an equivalence relation...

The binary relation R = {(1, 1)}, (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4) } on the set A = { 1, 2, 3, 4} is

The number of equivalence relations on the set $$\left\{ {1,2,3,4} \right\}$$ is

Suppose $$X$$ and $$Y$$ are sets and $$\left| X \right|$$ and $$\left| Y \right|$$ are their respective cardinalities. It is given that there are exactly 97 functions from $$X$$ to $$Y$$. From this one can conclude that

Let R denote the set of real numbers. Let f: $$R\,x\,R \to \,R\,x\,R\,$$ be a bijective function defined by f (x, y ) = (x + y, x - y). The inverse function of f is given by

Let R be a non-emply relation on a collection of sets defined by $${A^R}\,B $$ if and only if $$A\, \cap \,B\, = \,\phi $$. Then, (pick the true statement)

Which of the following statements is false?

Which one of the following is false?

Let $$X$$ $$X = \left\{ {2,3,6,12,24} \right\}$$. Let $$ \le $$ the partial order defined by $$x \le y$$ if $$x$$ divides $$y$$. The number of edges in the Hasse diagram of $$\left( {X, \le } \right)$$ is

Let $$R$$ be a symmetric and transitive relation on a set $$A$$. Then

Let $${G_1}$$ and $${G_2}$$ be subgroups of a group $$G$$. (a) Show that $${G_1}\, \cap \,{G_2}$$ is also a subgroup of $$G$$. (b) $${\rm I}$$s $${G_1}\, \cup \,{G_2}$$ always a subgroup of $$G$$?

Let A be the set of all nonsingular matrices over real numbers and let * be the matrix multiplication operator. Then

Some group (G, o) is known to be abelian. Then, which one of the following is true for G?

(a) If G is a group of even order, then show that there exists an element $$a \ne e$$, the identifier $$g$$, such that $${a^2} = e$$ (b) Consider the set of integers $$\left\{ {1,2,3,4,6,8,12,24} \right\}$$ together with...

The transitive closure of the relation $$\left\{ {\left( {1,2} \right)\left( {2,3} \right)\left( {3,4} \right)\left( {5,4} \right)} \right\}$$ on the set $$A = \left\{ {1,2,3,4,5} \right\}$$ is ________ .

State whether the following statement are TRUE or FALSE: (a) The union of two equivalence relations is also an equivalence relation.