Question

prove (a) A(+B)=AB & (b) A+B=A+B both algebraically and using the truth table.

Answer 1

A(+B)
=A+AB (distributivity)
=φ+AB (complement law)
=AB

A+B
=A+AB + B (absorption law)
=A + AB+B (associative law)
=A + (A+)B (distributive law)
=A + EB (complement law)
=A + B (identity law)

For the truth table, I'll do the first one, the second one is the same process.

A B +B A(+B) AB
T T F T T T
T F F F F F
F T T T F F
F F T T F F
We see that the truth table for A(+B) is identical to that of AB, therefore A(+B)≡AB