Relations and Functions · Mathematics · Class 12

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x)=x^2-4 x+5$ is

Let $$A=\{3,5\}$$. Then number of reflexive relations of $$A$$ is:

(a) Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let $L$ be the set of all rail lines on the railway track and $R$ be the relation on $L$ defined by

$R=\left\{l_1, l_2\right): l_1$ is parallel to $\left.l_2\right\}$

On the basis of the above information, answer the following questions:

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation R is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the equation $y=3 x+2$, then find the set of rail lines in R related to it.

OR

(b) Let $S$ be the relation defined by $S=\left\{\left(l_1, l_2\right): l_1\right.$ is perpendicular to $l_2$ \} check whether the relation $S$ is symmetric and transitive.

Case Study I

An organization conducted bike race under two different categories-Boys and Girls. There were 28 participants in all. Among all of them, finally three from category 1 and two from category 2 were selected for the final race. Ravi forms two sets $$B$$ and $$G$$ with these participants for his college project.

let $$B=\left\{b_1, b_2, b_3\right\}$$ and $$G=\left\{g_1, g_2\right\}$$, where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions:

(i) How many relations are possible from $$B$$ to $$G$$ ?

(ii) Among all the possible relations from $$B$$ to $$G$$, how many functions can be formed from $$B$$ to $$\mathrm{G}$$ ?

(iii) Let $$\mathrm{R}: \mathrm{B} \rightarrow \mathrm{B}$$ be defined by $$R=\{(x, y): x$$ and $$y$$ are students of the same sex $$\}$$. Check if $$R$$ is an equivalence relation.

OR

(iii) A function $$f: \mathrm{B} \rightarrow \mathrm{G}$$ be defined by $$f=\left\{\left(b_1, g_1\right)\right.$$, $$\left.\left(b_2, g_2\right),\left(b_3, g_1\right)\right\}$$.

Check if $$f$$ is bijective. Justify your answer.