GATE CSE 1996 | Set Theory & Algebra Question 89 | Discrete Mathematics

GATE CSE 1996

MCQ (Single Correct Answer)

-0.6

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

$${f^{ - 1}}\,(x,\,y)\, = \,\left( {{1 \over {x\, + \,y}},\,{1 \over {x\, - \,y}}} \right)$$

$${f^{ - 1}}\,(x,\,y)\, = \,\,(x\, - \,y,\,\,x\, + y)$$

$${f^{ - 1}}\,(x,\,y)\, = \,\left( {{{x\, + \,y} \over 2},\,{{x\, - \,y} \over 2}} \right)$$

$${f^{ - 1}}\,(x,\,y)\, = \,(2\,(x\, - \,y),\,2\,(x\, + y))$$

GATE CSE 1996

MCQ (Single Correct Answer)

-0.6

Which one of the following is false?

The set of all bijective functions on a finite set forms a group under function composition.

The set {1, 2, ..., p - 1} forms a group under multiplication mod p where p is a prime number.

The set of all strings over a finite alphabet $$\sum $$ forms a group under concatenation.

A subset $$s\, \ne \,\phi $$ of G is a subgroup of the group if and only if for any pair of elements $$a,\,\,b\,\, \in \,\,s,\,\,a\,\,*\,\,{b^{ - 1}}\,\, \in \,s$$.

GATE CSE 1996

MCQ (Single Correct Answer)

-0.6

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)

R is reflexive and transitive

R is symmetric and not transitive

R is an equivalence relation

R is not reflexive and not symmetric

GATE CSE 1995

MCQ (Single Correct Answer)

-0.6

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

A is closed under * but $$ < A,\,* > $$ is not a semigroup.

$$ < A,\,* > $$ is a semigroup but not a monoid.

$$ < A,\,* > $$ is a monoid but not a group.

$$ < A,\,* > $$ is a group but not an abelian group