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Let L be the set of all straight lines in the plane. Define a relation P on L as follows For all l_i,
l_2 epsilon L, l_1 Pl_2 iff l_1 is parallel to l_2. Prove that P is an equivalence relation on L and
describe the equivalence classes of P.
Solution
1.
For any line, l
lPl ie l is parallel to itself
Hence, P is reflexive
2.
For any two lines: l,r
lPr means rPl ie l parallel to r means r parallel to l
Hence, P is symmetric
3.
Let, l1 P l2, l2 P l3
Hence, l1 is parallel to l2 and l2 is parallel to l3
Hence, l1 is parallel to l3
Hence,l1 P l3
Hence, R is transitive.
Consider the line: y=mx . All lines of slope m will be parallel to this
So equivalence classes are :y=mx . For each different real number, m we have a different
equivalence class.

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Let L be the set of all straight lines in the plane. Define a relatio.pdf

  • 1. Let L be the set of all straight lines in the plane. Define a relation P on L as follows For all l_i, l_2 epsilon L, l_1 Pl_2 iff l_1 is parallel to l_2. Prove that P is an equivalence relation on L and describe the equivalence classes of P. Solution 1. For any line, l lPl ie l is parallel to itself Hence, P is reflexive 2. For any two lines: l,r lPr means rPl ie l parallel to r means r parallel to l Hence, P is symmetric 3. Let, l1 P l2, l2 P l3 Hence, l1 is parallel to l2 and l2 is parallel to l3 Hence, l1 is parallel to l3 Hence,l1 P l3 Hence, R is transitive. Consider the line: y=mx . All lines of slope m will be parallel to this So equivalence classes are :y=mx . For each different real number, m we have a different equivalence class.