Discreet Math set computation and proofs Find necessary and

Discreet Math, set computation and proofs Find necessary and sufficient condition (s) on sets X and Y for the statement. (X times Y) n (y times X) {} to be true and prove your result.

Solution

(X x Y) represents the cartesian product of X and Y whereas (Y x X) represnts the cartesian prouct of Y and X

Necessary and sufficient condition on sets X and Y

At least one element must be same on the set X and set Y, otherwise the quantity will be equal to the null set since there will no be common elements in (X x Y) and (Y x X)

Let the set X be {a,b,c} and Y be {d,e,f}

(X x Y) = {(a,d),(a,e),(a,f),(b,d),(b,e),(b,f),(c,d),(c,e),(c,f)}

(Y x x) = {(d,a),(e,a),(f,a),(d,b),(e,b),(f,b),(d,c),(e,c),(f,c)}

the intersection of these two sets will be null since there is no common elements

now let us suppose X = {a,b}, Y = {a,c}

X x Y = {(a,a),(a,c),(b,a),(b,c)}

Y x X = {(a,a),(a,b),(c,a),(c,b)}

Hence there will be one element in the intersection is equal to {a,a} which is not null element or phi