3. Show me lhe following relations on N are equivalence relations :
(i) R = {(x, y) e N ´N : x2 + y = y2 + x}
Answers
Answer:
Reflexive: Suppose x ∈ R. Then x − x = 0, which is an integer. Thus, xRx.
II. Symmetric: Suppose x, y ∈ R and xRy. Then x − y is an integer. Since
y − x = −(x − y), y − x is also an integer. Thus, yRx.
III. Suppose x, y ∈ R, xRy and yRz. Then x − y and y − z are integers. Thus,
the sum (x − y) + (y − z) = x − z is also an integer, and so xRz.
Thus, R is an equivalence relation on R.
Discussion
Example 3.2.2. Let R be the relation on the set of real numbers R in Example
1. Prove that if xRx0 and yRy0
, then (x + y)R(x
0 + y
0
).
Proof. Suppose xRx0 and yRy0
. In order to show that (x+y)R(x
0+y
0
), we must
show that (x + y) − (x
0 + y
0
) is an integer. Since
(x + y) − (x
0 + y
0
) = (x − x
0
) + (y − y
0
),
Answer:
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Step-by-step explanation:
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