Exercise 13 Let X Y and Z be sets Prove XY Z XYXZ XY Z XYX

Exercise 1.3. Let X, Y and Z be sets. Prove X(Y Z) = (XY)(XZ)

X(Y Z) = (XY)(XZ).

1) X(Y Z) = (XY)(XZ)

let a X(Y Z). Then a X and a Y Z.

Thus a X and a Y or a Z.

Hence a X and a Y or a X and a Z.

In other words, we have a X Y or a X Z.

Then, a (XY)(XZ)

Conversely, suppose a (XY)(XZ)

Then, a X Y and a X Z.

If a X, then a (Y Z).

If a X, then we must have a Y or a Z.

Hence, a Y Z and so a X (Y Z).

Therefore,

X(Y Z) = (XY)(XZ)

2) X(Y Z) = (XY)(XZ)

Suppose a X(Y Z)

Then a X or a Y Z.

If a X, then a belongs to both X Y and X Z;

Hence, a (X Y) (X Z).

If a Y Z, then a Y and a Z;

Hence, we also have a (XY) (X Z).

Conversely, suppose a (X Y) (X Z).

Then, a X Y and a XZ.

If a X, then a X (Y Z).

If a X, then we must have a Y and a Z.

Hence, a Y Z and so a X (Y Z).

Therefore,

X(Y Z) = (XY)(XZ)