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Let L be the set of all lines in X Y=...

Let `L` be the set of all lines in `X Y=p l a n e` and `R` be the relation in `L` defined as `R={(L_1,L_2): L_1` is parallel to `L_2}dot` Show that `R` is an equivalence relation. Find the set of all lines related to the line `y=2x+4.`

Text Solution

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`R = {(L1, L2): L1` is parallel to `L2}`.
`R` is reflexive as any line `L1` is parallel to itself, `(L1, L1) in R`.
Now,
Let `(L1, L2) in R`.
`=> L1` is parallel to `L2` and `L2` is parallel to ` L1`
`=> (L2, L1) in R`.
`:. R` is symmetric.
Now,
...
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