Select the relations that are functions.

a. {(a, 1), (b, 1), (c, 1)}
b. {(a, a), (a, b), (a, c)}
c. {(1, a), (2, a), (3, a)}
d. {(a, a), (b, b), (c, c)}

The correct options are A, B and C.

Step-by-step explanation:

A relation is called a function if there exist a unique output for each input. In other words there exist a unique value of y for each value of x.

In relation A,

{(a, 1), (b, 1), (c, 1)}

In the above relation, for each value of x, we have a unique value of y, therefore this relation is a function and option A is correct.

In relation B,

{(a, a), (a, b), (a, c)}

In the above relation, for each value of x, we have same value of y, therefore this relation is not a function and option B is incorrect.

In relation C,

{(1, a), (2, a), (3, a)}

In the above relation, for each value of x, we have a unique value of y, therefore this relation is a function and option C is correct.

In relation D,

{(a, a), (b, b), (c, c)}

In the above relation, for each value of x, we have a unique value of y, therefore this relation is a function and option D is correct.

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.