Sets - Problem Solving

If $A = \{ 1,2,3,4 \}, B = \{ 4,5,6,7 \},$ determine the following sets:

(i) $A \cap B$
(ii) $A \cup B$
(iii) $A \backslash B$

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(i) By definition, $\cap$ tells us that we want to find the common elements between the two sets. In this case, it is 4 only. Thus $A \cap B = \{ 4 \}$.

(ii) By definition, $\cup$ tells us that we want to combine all the elements between the two sets. In this case, it is $A\cup B = \{1,2,3,4,5,6,7 \}$.

(iii) By definition, $\backslash$ tells us that we want to look for elements in the former set in that doesn't appear in the latter set. So $A \backslash B = \{1,2,3\}$. $_\square$

Consider the same example above. If the element $4$ is removed from the set $B$, solve for (i), (ii), (iii) as well.

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(i) Since there is no common elements in sets $A$ and $B$, then $A \cap B = \phi$ or $A \cap B = \{ \}$.

(ii) Because the element $4$ is no longer repeated, then $A \cup B$ remains the same.

(iii) Since $A$ and $B$ no longer share any common element, $A\backslash B$ is simply equals to set $A$, which is $\{1,2,3,4 \}$. $_\square$