1
GATE CSE 1998
Subjective
+2
-0
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) Draw the graph of the above equivalence relation.

2
GATE CSE 1996
+2
-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)
A
R is reflexive and transitive
B
R is symmetric and not transitive
C
R is an equivalence relation
D
R is not reflexive and not symmetric
3
GATE CSE 1996
+2
-0.6
Which one of the following is false?
A
The set of all bijective functions on a finite set forms a group under function composition.
B
The set {1, 2, ..., p - 1} forms a group under multiplication mod p where p is a prime number.
C
The set of all strings over a finite alphabet $$\sum$$ forms a group under concatenation.
D
A subset $$s\, \ne \,\phi$$ of G is a subgroup of the group if and only if for any pair of elements $$a,\,\,b\,\, \in \,\,s,\,\,a\,\,*\,\,{b^{ - 1}}\,\, \in \,s$$.
4
GATE CSE 1996
+2
-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
A
$${f^{ - 1}}\,(x,\,y)\, = \,\left( {{1 \over {x\, + \,y}},\,{1 \over {x\, - \,y}}} \right)$$
B
$${f^{ - 1}}\,(x,\,y)\, = \,\,(x\, - \,y,\,\,x\, + y)$$
C
$${f^{ - 1}}\,(x,\,y)\, = \,\left( {{{x\, + \,y} \over 2},\,{{x\, - \,y} \over 2}} \right)$$
D
$${f^{ - 1}}\,(x,\,y)\, = \,(2\,(x\, - \,y),\,2\,(x\, + y))$$
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