Question

A city’s traffic management department is planning to optimize traffic flow by analyzing the connectivity between various traffic signals. The city has five major spots labelled A, B, C, D, and E. The department has collected the following data regarding one-way traffic flow between spots: Traffic flows from A to B, A to C and A to D.  Traffic flows from B to C and B to E.  Traffic flows from C to E.  Traffic flows from D to E and D to C.

The department wants to represent and analyze this data using relations and functions. Use the given data to answer the following questions:

1. Is the traffic flow reflexive? Justify.  (1)
2. Is the traffic flow transitive? Justify.  (1)
3. 1. Represent the relation describing the traffic flow as a set of ordered pairs. Also state the domain and range of the relation.  (2)
      OR
   2. Does the traffic flow represent a function? Justify your answer.

Answer 1

Traffic flow is a relation which looks like

R = {(A, B), (A, C), (A, D), (B, C), (B, E), (C, E), (D, E), (D, C)}

I. If the relation is reflexive, then (x, x) ∈ R for every x ∈ A, i.e., (A, B, C, D, E)

Since (A, A) ∉ R

∴ R is not reflexive.

II. If (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R

Here, (A, C) ∈ R and (C, E) ∈ R but (A, E) ∉ R.

∴ R is not transitive.

III. A.

Domain = Set of all first elements of ordered pair

= {A, B, C, D}

Range = Set of all second elements of ordered pair

= {B, C, D, E}

OR

III. B. Function is a special type of relation where

1. Every element will have an image.
2. Every element will have only one image

Now, R = {(A, B), (A, C), (A, D), (B, C), (B, E), (C, E), (D, E), (D, C)}

Here, A has 3 images, B has 2 images, D has 2 images and E has no images.

Thus, the traffic flow is not a function.