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Question
Let L be the set of all lines in the 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.
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Solution
(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 the 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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