Let the universe of discourse U be the set of all people and

Let the universe of discourse, U, be the set of all people, and let M(x, y) be \"x is the mother of y.\"

Translate the following statement into logical notation using quantifiers and the proposition M(x, y) over U: \"Everyone has a grandmother,\"

Solution

for everybody has a mother , wecan write,

yx:M(x,y) for all y, there exist a x such that x is the mother of y.

now we have to move forward to mother \'s mother.

for all z, there exist a x and y such that y is mother of x and z is the mother of y.

zyx:M(x,y)M(y,z)

in the limitation of question we have to assume that grandmother here is referred to mother\'s mother.