For all sets X,Y and Z, prove that if X⊆Y, then (i) X ∩ Z ⊆ Y ∩ Z (ii) X ∪ Z ⊆ Y ∪ Z The problem should be solved in indirect proof way. Thanks!
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(i) Observe that since $\displaystyle X\subseteq Y$ this means $\displaystyle X = X\cap Y$. Now by using the above observation you want to show the implication: $\displaystyle x\in X\cap Z \Rightarrow x\in Y\cap Z$ (ii) Same trick.
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