Question

(i) Show that if A ⊆ C and B ⊆ D, then A×B ⊆ C ×D
(ii) Let A ={a , b , c ,d}and B ={y , z}. Find (a) A×B (b) B×A

Answer 1

(i) Take $(x,y)\in A\times B$. Then $x\in A$ and $y\in B$, hence by $A\subset C$ and $B\subset D$ we have $x\in C$ and $y\in D.$ This implies that $(x,y)\in C\times D$, which gives $A\times B\subset C\times D$.

(ii) By definition, $A\times B=\{(a,y),(a,z),(b,y),(b,z),(c,y),(c,z),(d,y),(d,z)\}$ and $B\times A=\{(y,a),(z,a),(y,b),(z,b),(y,c),(z,c),(y,d),(z,d)\}$