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 \(\{1,2,3,4 \} \). \(_\square\)