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Question
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.
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Solution
We observe the following properties of R.
Reflexivity: Let L1 be an arbitrary element of the set L. Then,
L1 ∈ L
⇒ L1 is parallel to L1 [Every line is parallel to itself]
⇒ (L1, L1) ∈ R for all L1 ∈ L
So, R is reflexive on L.
Symmetry : Let (L1, L2) ∈ R
⇒ L1 is parallel to L2
⇒ L2 is parallel to L1
⇒ (L2, L1) ∈ R for all L1 and L2 ∈ L
So, R is symmetric on L.
Transitivity : Let (L1, L2) and (L2, L3) ∈ R
⇒ L1 is parallel to L2 and L2 is parallel to L3
⇒ L1, L2 and L3 are all parallel to each other
⇒L1 is parallel to L3
⇒(L1, L3) ∈ R
So, R is transitive on L.
Hence, R is an equivalence relation on L.
Set of all the lines related to y = 2x+4
= L' = {(x, y) : y = 2x+c, where c∈ R}
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