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iamsmooth
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Homework Statement
Prove that [tex]\forall[/tex] sets [tex] A, B, C [/tex], if [tex]B\subseteq C, [/tex] then [tex] A \times B \subseteq A \times C[/tex]
Homework Equations
The Attempt at a Solution
Haven't done set theory proofs in a while. Does this suffice in proving the statement?:
Let [tex] x \in B [/tex] be arbitrary. Assuming [tex]B \subseteq C[/tex] is true, we know that [tex]x \in C[/tex]. Since [tex]x \in B[/tex], we know that [tex]A \times B[/tex] will produce an ordered pair (a,x) where a is an arbitrary element of A. Since [tex]x \in C[/tex], we know that [tex]A \times C[/tex] will produce the same ordered pair (a,x).
Therefore by definition of subsets, [tex]A \times B \subseteq A \times C[/tex]
QEDThanks for your help.
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