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.
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 L2.
⇒ L2 is parallel to L1.
⇒ (L2, L1 ) ∈ R
∴ R is symmetric.
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 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.
A compound (A) with molecular formula $C_4H_9I$ which is a primary alkyl halide, reacts with alcoholic KOH to give compound (B). Compound (B) reacts with HI to give (C) which is an isomer of (A). When (A) reacts with Na metal in the presence of dry ether, it gives a compound (D), C8H18, which is different from the compound formed when n-butyl iodide reacts with sodium. Write the structures of A, (B), (C) and (D) when (A) reacts with alcoholic KOH.
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Every element of set A is related to itself only then the relation is identity relation.
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.