Consider the statement xy 2x 3y 1 Is this true or false if

Consider the statement x!y (2x – 3y = 1).

Is this true or false if the universe of discourse is x, y ? Either provide a proof to show that this is true or provide a counterexample to show that this is false.

Is this true or false if the universe of discourse is x, y ? Either provide a proof to show that this is true or provide a counterexample to show that this is false.

Solution

Q is set of all rational numbers. and 2x-3y =1 which assumes all real values of y for all x i.e. domain is R and range also R. Since Q is a subset of R (real numbers) hence the statement is true.

But for x ,y belonging to N (natural numbers ) this is not true. since for x =1 , y = (2*1-1)/3 = 1/3 which is not a natural number. also for y=2, x =(1+3*2)/2= 7/2 which is also not a natural number. hene statement is false.