1
GATE CSE 1998
Subjective
+2
-0
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 for every a, b, c in A
a * b * c = a * c

2
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.

3
GATE CSE 1996
MCQ (Single Correct Answer)
+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
4
GATE CSE 1996
MCQ (Single Correct Answer)
+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 $$a,\,\,b\,\, \in \,\,s,\,\,a\,\,*\,\,{b^{ - 1}}\,\, \in \,s$$.
GATE CSE Subjects
Software Engineering
Web Technologies
EXAM MAP
Medical
NEET
Graduate Aptitude Test in Engineering
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CBSE
Class 12