Question

Given A, B and C are sets, prove or disprove the following:
(a) A subset B subset C <==> AxB subset BxC

Answer 1

\[A \subseteq B \subseteq t C <=> A \times B \subseteq B \times C\]

Answer 2

typing error..no t in this expression

Answer 3

can probably do this working directly from the definitions

Answer 4

first show \[A \subseteq B \subseteq C \implies A \times B \subseteq B \times C\]

Answer 5

if and only if

Answer 6

yeah you have to go both ways. first the implication above. you assume
\[A \subseteq B \subseteq C \] and show
\[A \times B \subseteq B \times C\]

Answer 7

by definition
\[A\times B=\{(a,b)|:a\in A, b\in B\}\]
let \((a,b)\in A\times B\) then since
\(A \subseteq B, a\in B\) and since \(B \subseteq C, b\in C \) therefore \((a,b)\in B\times C\)

Answer 8

what about left side to right side ? how to prove it..