Let L be the set of all lines in
XY plane and R be the relation in L defined as R = {(L1,
L2): L1 is parallel
to L2}. Show that R is an equivalence relation.
Find the set of all lines related to the line y = 2x +
4.
Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
R = {(L1,
L2): L1 is parallel to L2}
R is
reflexive as any line L1 is parallel to itself
i.e., (L1, L1) ∈
R.
Now,
Let (L1,
L2) ∈ R.
⇒
L1 is parallel to L2.
⇒
L2 is parallel to L1.
⇒
(L2, L1) ∈
R
∴ R
is symmetric.
Now,
Let (L1,
L2), (L2, L3)
∈R.
⇒
L1 is parallel to L2. Also, L2
is parallel to L3.
⇒
L1 is parallel to L3.
∴R
is transitive.
Hence, R
is an equivalence relation.
The set of
all lines related to the line y = 2x + 4 is the set of
all lines that are parallel to the line y = 2x + 4.
Slope of
line y = 2x + 4 is m = 2
It is
known that parallel lines have the same slopes.
The line
parallel to the given line is of the form y = 2x + c,
where c ∈R.
Hence, the
set of all lines related to the given line is given by y = 2x
+ c, where c ∈
R.
R = {(L1, L2): L1 is parallel to L2}
R is reflexive as any line L1 is parallel to itself i.e., (L1, L1) ∈ R.
Now,
Let (L1, L2) ∈ R.
⇒ L1 is parallel to L2.
⇒ L2 is parallel to L1.
⇒ (L2, L1) ∈ R
∴ R is symmetric.
Now,
Let (L1, L2), (L2, L3) ∈R.
⇒ L1 is parallel to L2. Also, L2 is parallel to L3.
⇒ L1 is parallel to L3.
∴R is transitive.
Hence, R is an equivalence relation.
The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.
Slope of line y = 2x + 4 is m = 2
It is known that parallel lines have the same slopes.
The line parallel to the given line is of the form y = 2x + c, where c ∈R.
Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.
