Respuesta :
For part (a) and (b) suposse [tex]A\subset B[/tex].
To prove part (a) observe that we already have that [tex]B\subset A\cup B[/tex]. So we will prove that [tex]A\cup B \subset B[/tex]. Let [tex]x\in A\cup B[/tex], then [tex]x\in A[/tex] or [tex]x\in B[/tex]. If [tex]x\in B[/tex] we finish the proof, and if [tex]x\in A[/tex] implies [tex]x\in B[/tex] because we assume [tex]A\subset B[/tex], and the proof is complete.
For part (b) we always have [tex]A\cap B\subset A[/tex]. We finish the proof showing [tex]A\subset A\cap B[/tex]. Let [tex]x\in A[/tex], then [tex]x\in B[/tex] by the asumption that [tex]A\subset B[/tex]. So, we have both [tex]x\in A[/tex] and [tex]x\in B[/tex], that implies [tex]x\in A\cap B[/tex]. Therfore [tex]A\subset A\cap B[/tex], which completes the proof.
