1
GATE CSE 1998
+1
-0.3
A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is
A
1/6
B
3/8
C
1/8
D
1/2
2
GATE CSE 1998
+2
-0.6
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
A
Reflexive, symmetric and transitive
B
Neither reflexive, nor irreflexive but transitive
C
Irreflexive, symmetric and transitive
D
Irreflexive and antisymmetric
3
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

4
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.

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