In a test generation process for a combinational circuit six tests, t$1,$ t$2,$ t$3,$ t$4,$ t$5,$ t$6$ are generated to cover a set of given faults. Later it is discovered that we are interested only in a subset of the faults and the subset consists of eight faults, f$1,$ f$2,$ f$3,$ f$4,$ f$5,$ f$6,$ f$7,$ f$8.$

Though simulating the six tests for each of the faults $($without fault dropping$)$ we find the detection capability of each test as given below.

The test t$1$ can detect faults f$3$ and f$5$

The test t$2$ can detect faults f$2$ and f$7$

The test t$3$ can detect faults f$2,$ f$3,$ and f$7$

The test t$4$ can detect faults f$1,$ f$2,$ and f$7$

The test t$5$ can detect faults f$4,$ and f$6$

The test t$6$ can detect faults f$1,$ f$4,$ f$6$ and f$8.$

Find a smallest set of tests that can detect all eight faults. You must show your work to prove that the set obtained by you is the smallest set.