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Joint Entrance Examination
Mathematics
Sets, Relations and Functions
Question 91
Question 91 of 161
Easy
MCQ
The relation
R
=
{
(
x
,
y
)
:
x
,
y
∈
Z
,
x
+
y
is even
}
R = \{(x,y) : x,y\in\mathbb{Z},\ x+y \text{ is even}\}
R
=
{(
x
,
y
)
:
x
,
y
∈
Z
,
x
+
y
is even
}
is:
A
reflexive and transitive but not symmetric
B
reflexive and symmetric but not transitive
C
an equivalence relation
D
symmetric and transitive but not reflexive
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Official solution
- Reflexive:
x
+
x
=
2
x
x+x=2x
x
+
x
=
2
x
is always even. - Symmetric:
x
+
y
x+y
x
+
y
even
⟺
\iff
⟺
y
+
x
y+x
y
+
x
even. - Transitive: if
x
+
y
x+y
x
+
y
and
y
+
z
y+z
y
+
z
are both even, then
(
x
+
y
)
+
(
y
+
z
)
=
x
+
z
+
2
y
(x+y)+(y+z) = x+z+2y
(
x
+
y
)
+
(
y
+
z
)
=
x
+
z
+
2
y
is even, and since
2
y
2y
2
y
is even,
x
+
z
x+z
x
+
z
must be even. So
R
R
R
is an equivalence relation.
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