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If you have a good understanding of the logic gate rules, it is possible to work out a simpler solution just by examining the truth table and thinking hard. But this is a rather hit and miss approach and Karnaugh maps and Boolean algebra can be faster and more reliable methods but neither is easy.
Boolean algebra can be used to simplify expressions/circuits to generate a more optimal solution.
Everyone
De Morgan Boolean Identities
De Morgan

A + B = A . B

A . B = A + B
Gate Behaviour

AND GATE

A . 1 = A The gate is open.

A . 0 = 0 The gate is closed  output low.

OR GATE

A + 1 = 1 The gate is closed  output high.

A + 0 = A The gate is open.
Commutative

When using AND and OR, it makes no difference if the inputs are swapped.

A . B = B . A

A + B = B + A
Associative

The presence or absence of brackets makes no difference.

A . (B . C) = (A . B) . C = A . B . C

A + (B + C) = (A + B) + C = A + B + C
Distributive

A . (B + C) = A . B + A . C

(A + B) . (A + C) = A + (B . C)
Identity

A + A = A

A . A = A

A . B + A . B = A

(A + B) . ( A + B) = A
Redundancy
In all these expressions, one of the items has no effect and can be removed.

A . (A + B) = A

A + A . B = A

0 . A = 0

0 + A = A

1 . A = A

1 + A = 1

A . A = 0

A + A = 1

A . (A + B) = A . B

A + A . B = A + B
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