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let l be the set of all lines in xy plane and r be...

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.

Answer:

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 L_2.

⇒ L₂ is parallel to L_1.

⇒ (L₂, L₁) ∈ R

∴ R is symmetric.

Now,

Let (L₁, L₂), (L₂, L₃) ∈R.

⇒ L₁ is parallel to L₂. Also, L₂ is parallel to L_3.

⇒ L₁ is parallel to L_3.

∴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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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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