Functions-7

If a, b ∈ {1, 2, 3, 4}, then which of the following are functions in the given set?
(a) f1 = {(x,y) : y = x + 1} (b) f2 = {(x,y) : x + y > 4}
(c) f3 = {(x,y) : y < x} (d) f4 = {(x,y) : x + y = 5}
Also, in case of a function give its range.

Solution: If we express f1, f2, f3 and f4 as set of ordered pairs, then we have

f1 = {(1,2), (2,3), (3,4)}

f2 = {(1,4), (4,1), (2,3), (3,2), (2,4), (4,2), (3,4), (4,3)}

f3 {(2,1), (3,1), (4,1), (3,2), (4,2), (4,3)} and

f4 = {(1,4), (2,3), (3,2), (4,1)}

(a) We have: f1 = {(1,2), (2,3), (3,4)}.

We observe that an element 4 of the given set has not appeared in the first place of any ordered pair of f1. So,

f1 is not a function from the given set to itself.

(b) We have f2 = {(1,4), (4,1), (2,3), (3,2), (2,4), (4,2), (3,4), (4,3)}

We observe that 2, 3, 4 have appeared more than once as first components of the ordered pairs in f2. So, f2 is

not a function.

(c) We have f3 = {(2,1), (3,1), (4,1), (3,2), (4,2), (4,3)}

We observe that 3 and 4 have appeared more than once as first components of the ordered pairs in f3. So, f3 is

not a function.

(d) We have f4 = {(1,4), (2,3), (3,2), (4,1)}

We observe that each element of the given set has appeared as first components in one and only one ordered pair of f4. So,

f4 is a function in the given set. In this case,

Range of f {1, 2, 3, 4}