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प्रश्न
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.
उत्तर
(i) Reflexive
R = {(L1, L2): L1 is parallel to L2}
R is reflexive as any line L1 is parallel to itself, i.e., (L1, L1) ∈ R.
∴ R is reflexive
(ii) Symmetric
Now, let (L1, L2) ∈ R
⇒ L1 is parallel to L2
⇒ L2 is parallel to L1
⇒ (L2, L1) ∈ R
∴ R is symmetric
(iii) Transitive
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
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