As you may have seen in CSC 28, if f is a binary relation between sets A and B w

Question

As you may have seen in CSC 28, if f is a binary relation between sets A and B with |A| ordered pairs in the relation and every element of A occurs exactly once as a first element in an ordered pair, then the relation can be viewed as a function f: A rightarrow B. a) Let A = {1, 2, 3} and B = {a, b, c}. Is f = {(1, a), (2, b)} an invertible function? If so, what is its inverse (given as a set of ordered pairs). If not, explain in one short sentence. b) Let A = {1, 2, 3} and B = {a, b, c}. Is f = {(1, a), (2, b), (3, b)} an invertible function? If so, what is its inverse (given as a set of ordered pairs). If not, explain in one short sentence. c) Let A = {1, 2, 3} and B = {a, b, c}. Is f = {(1, a), (2, b), (3, c)} an invertible function? If so, what is its inverse (given as a set of ordered pairs). If not, explain in one short sentence.

Explanation / Answer

A function f:A->B is invertible, only when it has the property that:

For every y which is in the set Y, there should be only one x in the set X, so that f(x) = y.

a) Here, f is not invertible. Because there exists the element 'c' in B, for which there is no x in X, such that f(x) = c.

b) Here, f is not invertible. Two reasons: there exists the element 'c' in B, for which there is no x in X, such that f(x) = c. Also, for the element 'b' in B, we have two elements in X (2 and 3), such that f(x) = b [i.e. f(2) = f(3) = 3].

c) Here f is invertible, because for all the elements y in the set Y, there exists only one x in X such that f(x) = y. The inverese function g can be written as, f = {(a,1),(b,2),(c,3)}. Clearly, g(a) = 1 [whereas, f(1) was a], g(b) = 2 [whereas, f(2) was b] and g(c) = 3 [whereas, f(3) was c].

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