1
GATE CSE 2001
+1
-0.3
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 over the set of integers
$${R_3}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,a.b > 0$$ over the set of non-zero rational numbers
$${R_4}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left| {a - b} \right| \le 2$$ over the set of natural numbers

Which of the following statements is correct?

A
$${R_1}$$ and $${R_2}$$ are equivalence relations, $${R_3}$$ and $${R_4}$$ are not
B
$${R_1}$$ and $${R_3}$$ are equivalence relations, $${R_2}$$ and $${R_4}$$ are not
C
$${R_1}$$ and $${R_4}$$ are equivalence relations, $${R_2}$$ $${R_3}$$ are not
D
$${R_1}$$, $${R_2}$$, $${R_3}$$ and $${R_4}$$ are all equivalence relations
2
GATE CSE 1999
+1
-0.3
The number of binary relations on a set with $$n$$ elements is:
A
$${n^2}$$
B
$${2^n}$$
C
$$2{n^2}$$
D
None of the above
3
GATE CSE 1998
+1
-0.3
The number of functions from an $$m$$ element set to an $$n$$ element set is
A
$$m + n$$
B
$${m^n}$$
C
$${n^m}$$
D
$$m * n$$
4
GATE CSE 1998
+1
-0.3
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

Which of the following is correct?

A
both assertions are true
B
assertion
(i) is true but assertion (ii) is not true
C
assertion
(ii) is true but assertion (i) is not true
D
neither (i) nor (ii) is true
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