Let's simplify the given logical expression step by step:

Expression:

$(A \vee B)' \vee (A' \land B)$

1. Apply De Morgan's Law:
   The first part of the expression is $(A \vee B)'$, which can be simplified using De Morgan's Law. De Morgan's Law states: $(A \vee B)' = A' \land B'$ So, the expression becomes: $(A' \land B') \vee (A' \land B)$

2. Factor the Expression:
   Now, observe that both terms $(A' \land B')$ and $(A' \land B)$ have $A'$ in common. So, we can factor out $A'$: $A' \land (B' \vee B)$

3. Simplify the Expression Inside the Parentheses:
   Notice that $B' \vee B$ is a tautology, meaning it is always true (since either $B$ is true or $B$ is false). Therefore: $B' \vee B = \text{True}$ So, the expression simplifies to: $A' \land \text{True}$

4. Simplify the Final Expression:
   Since $A' \land \text{True} = A'$, the final simplified expression is: $A'$

Final Answer:

$A'$

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Would you like to see a breakdown of any specific part of the process? Or do you have more questions?

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Here are 5 related questions to explore:

1. What is the difference between De Morgan's laws and distributive laws in logic?
2. How can you prove that $A' \land \text{True} = A'$?
3. What is a tautology in logic? Can you give more examples?
4. What would happen if we had a different logical operator, like $A \land B$, instead of $\vee$?
5. How do we simplify logical expressions with more than two variables?

Tip: In logical expressions, look for common terms or patterns, such as tautologies ($B \vee B' = \text{True}$) and contradictions ($A \land A' = \text{False}$), which can help reduce complexity quickly.