Consider the following lines C 1 and C 2 : C 1 : y 3 = x 3 + 1 C 2 : y 3 = − x 3 + 1 Let C 3 be the straight line which is perpendicular to C 2 and whose y -intercept is 4 . Define S 1 to be the set of points of intersection of C 1 and C 2 S 2 to be the set of points of intersection of C 1 and C 3 S 3 to be the set of all the points of intersection of " C 1 and C 2 " or " C 2 and C 3 " or " C 1 and C 3 ". What is the cardinality of S 1 ∪ S 3 ?
Answers
Given : C1: y/3=x/3+1
C2: y/3=−x/3+1
C3 be the straight line which is perpendicular to C2 and whose y-intercept is 4.
S1 to be the set of points of intersection of C1 and C2 S2 to be the set of points of intersection of C1 and C3 S3 to be the set of all the points of intersection of "C1 and C2" or "C2 and C3" or "C1 and C3".
To find : cardinality of S 1 ∪ S 3
Solution:
C1: y/3=x/3+1
C2: y/3=−x/3+1
Adding both => 2y/3 = 2 => y = 3
3/3 = x/3 + 1 => x = 0
(0,3) is the intersection point of C1 and C2
S1 = "C1 and C2" = { (0, 3)}
C3 be the straight line which is perpendicular to C2
C2: y/3=−x/3+1
=> y = -x + 3 => slope = - 1
slope of perpendicular line to C2 = 1
C3 y = x + 4 as y-intercept is 4.
C1: y/3=x/3+1 => y = x + 1
C3 : y = x + 4
C1 and C3 are parallel lines so no intersection points
S2 = {} = ∅
C2: y = -x + 3
C3 y = x + 4
=> y = 7/2 , x = -1/2
C2 and C3 = {( -1/2 , 7/2)
S1 = C1 and C2 = { (0, 3)}
S2 = C1 and C3 = {} = ∅
C2 and C3 = {( -1/2 , 7/2)}
S3 = "C1 and C2" or "C2 and C3" or "C1 and C3".
=> S3 = { (0, 3) , ( -1/2 , 7/2) }
S1 = { (0, 3)} , S3 = { (0, 3) , ( -1/2 , 7/2) }
S 1 ∪ S 3 = { (0, 3) , ( -1/2 , 7/2) }
n (S 1 ∪ S 3 ) = 2
Hence cardinality of S 1 ∪ S 3 is 2
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