Question

Let $A=1,2,3,4,5$, $B=1,3,5,7,9,10,17,26,30.$ Which of the following sets of ordered pairs are (i) relations (ii) functions (iii) neither, from A to B
(a) $R_1$= {(1, 3), (3, 3), (5, 17), (1, 2)}
(b) $R_2$ = {(3, 7), (4, 5), ( 5, 30)}
(c) $R_3=(x,y):x\in A,y\in B,x+y=8$
(d) $R_4=(x,y):x\in A,y\in B,y=x^2+1$
(e) $R_5=(x,y):x\in A,y\in B,y=2x-1$

Answer 1

$R_{1}neqR$ as $(1,2)\notin A\times B,R_1\ne f$ as 2 or 4 $\in$ A has no image in B,

$R_2=R$ but $R_2\ne f$ as in $R_1$

$R_3=(1,7),(3,5),(5,3)$

$x+y=8,x\in A,y\in B$

$\therefore R_3$ = R being a subset of$A\times B$ but $R_3\ne f$ as in $R_1$

$R_4=(2,5),(3,10),(4,17),(5,26)$

$\because y=x^2+1,x\in A,y\in B$

$R_4=R$ but $R_4\ne f$ as in $R_1$

$R_5=(1,1),(2,3),(3,5),(4,7),(5,9)$

$y=2x-1$

$R_5$ = R and $R_5=f$as each element $\in A$ has a unique image in B.