Question

Consider the function \(f:\\{1,2,3,4,5,6,7\\} \rightarrow\\{0,1,2,3,4,5,6,7,8,9\\}\) given as $$f=\\{(1,3),(2,8),(3,3),(4,1),(5,2),(6,4),(7,6)\\}$$ Find: \(f(\\{1,2,3\\}), f(\\{4,5,6,7\\}), f(\varnothing), f^{-1}(\\{0,5,9\\})\) and \(f^{-1}(\\{0,3,5,9\\})\).

Short answer

\(f(\{1,2,3\}) = \{3, 8\}, f(\{4,5,6,7\}) = \{1, 2, 4, 6\}, f(\varnothing) = \varnothing, f^{-1}(\{0,5,9\}) = \varnothing, f^{-1}(\{0,3,5,9\}) = \{1, 3\}\).

Step-by-step solution

Find Images of Sets under function f

The function is given as specified pairs. Look at each pair to find the mapping for the necessary elements. For \(f(\{1,2,3\})\), look where the function maps 1, 2 and 3 in the given function. 1 maps to 3, 2 maps to 8 and 3 maps to 3, thus \(f(\{1,2,3\})\) = \{3, 8, 3\}, which we clean-up to \{3, 8\}, removing the repetition. Then for \(f(\{4,5,6,7\})\), 4 maps to 1, 5 maps to 2, 6 maps to 4 and 7 maps to 6, producing \(f(\{4,5,6,7\})\) = \{1, 2, 4, 6\}.

Find Image of an Empty Set under Function f

The function, f, cannot map anything from the empty set. This is because there are no elements in the empty set to map. So, \(f(\varnothing)\) = \varnothing.

Compute the Inverse Image under Function f

The inverse image computation involves finding the elements from the domain that map to the given element in the co-domain. So for \(f^{-1}(\{0,5,9\})\), look at the pairs given for the function f, there are no elements mapped to 0, 5, or 9, and thus \(f^{-1}(\{0,5,9\})\) = \varnothing. Now for \(f^{-1}(\{0,3,5,9\})\), the domain elements mapping to 3 belong to this set, specifically, 1 and 3. Thus, \(f^{-1}(\{0,3,5,9\})\) = \{1, 3\}.