The finite state machine given in figure below recognizes :

Key idea: each state encodes the parity (even or odd) of the number of a's and b's seen so far.

• Start state = (even a, even b).

• Top-right state = (odd a, even b).

• Bottom-left state = (even a, odd b).

• Bottom-right (accepting) = (odd a, odd b).

Transitions:

• Reading 'a' flips the a-parity and moves horizontally between top and bottom pairs.

• Reading 'b' flips the b-parity and moves vertically between left and right pairs.

Conclusion: the machine accepts exactly those strings where both the number of a's and the number of b's are odd (i.e., parity(a)=odd and parity(b)=odd).

Examples:

• Accepted: "ab" (start -> read a -> read b -> bottom-right accepting).

• Rejected: "a" (one a but zero b -> ends in non-accepting state).