Discrete Mathmatics

Exercise 2.6.8: Proving set identities with Cartesian products. About Use the following three definitions and the laws of logic to prove the two identities given below . Definition of Cartesian product (xyje A x B-, (XE A) ( B) . Definition of intersection: x A n B (xe A) (xe B) . Definition of union: x e A u B (xe A) v (xe B) (a) Ax (BUC) (Ax B) U (Ax C) Solution (xy) E A x (BU C) (XE A) ^ (y (BU C)), by the definition of Cartesian product (xe A) ((y e B) v (y e c), by the definition of union . (xe A) ^ (y e B)) v ((x E A) (y E C)), by the distributive law ((x .yje A B) V ((x,y) E A . C), by the definition of Cartesian product (x y)E (Ax B) U (Ax C), by the definition of union Feedback? Exercise 2.6.8: Proving set identities with Cartesian products. About Use the following three definitions and the laws of logic to prove the two identities given below . Definition of Cartesian product (xyje A x B-, (XE A) ( B) . Definition of intersection: x A n B (xe A) (xe B) . Definition of union: x e A u B (xe A) v (xe B) (a) Ax (BUC) (Ax B) U (Ax C) Solution (xy) E A x (BU C) (XE A) ^ (y (BU C)), by the definition of Cartesian product (xe A) ((y e B) v (y e c), by the definition of union . (xe A) ^ (y e B)) v ((x E A) (y E C)), by the distributive law ((x .yje A B) V ((x,y) E A . C), by the definition of Cartesian product (x y)E (Ax B) U (Ax C), by the definition of union Feedback