Question

find a natural bijection between the two sets X and y. where X is the set of all lines in R^2 parallel to the X axis and y=Rl​

Answer 1

Answer:

R={(L

1

,L

2

):L

1

is parallel to L

2

}

R is reflexive as any line L

1

is parallel to itself i.e., (L

1

,L

1

)∈R.

Now, let (L

1

,L

2

)∈R.

⇒L

1

is parallel to L

2

.

⇒L

2

is parallel to L

1

.

⇒(L

2

,L

1

)∈R

∴R is symmetric.

Now, let (L

1

,L

2

),(L

2

,L

3

)∈R.

⇒L

1

is parallel to L

2

. Also, L

2

is parallel to L

3

.

⇒L

1

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