# J K Flip Flop and Boolean Algebra

Y=K'J'Q+K'J+KJQ'

The output Y should be that of a JK flip-flop. That is: Y=JQ'+K'Q

I tried to solve the following way:

1. Y=K'J'Q+K'J+KJQ'

2. =K'(J'Q+J)+KJQ'

3. =JK'+K'Q+JKQ'

4. =J(K'+KQ')+K'Q

5. =J(K'+Q')+K'Q

6. =JK'+JQ'+K'Q

• Explain the last step, please? Oct 31, 2018 at 15:45
• @EugeneSh. Done Oct 31, 2018 at 15:49
• Try writing down the truth tables of the original function and the target one. If these are different, then you have missed something (or the question author did) Oct 31, 2018 at 15:54
• Also draw out the waveforms, with arrows to show what signals (some are anded) affect the outputs, at what times. Oct 31, 2018 at 16:23

Y = K'J'Q + K'J + KJQ'

The trick is to create two terms that are equivalent to the middle term, using the fact that X + X' = 1:

Y = J'K'Q + JK'(Q + Q') + JKQ'

Y = J'K'Q + JK'Q + JK'Q' + JKQ'

Now factor the pairs of terms:

Y = K'Q(J + J') + JQ'(K + K')

Simplify, again using the fact that X + X' = 1:

Y = K'Q + JQ'

This is really obvious if you draw the Karnaugh map for the function. The JK' term is redundant, given the other two terms.