Math Problem Statement
Show that xy+x'z+yz=xy+x'z
Solution
We will simplify step by step and show that it equals .
Expression:
Step 1: Factorize common terms
Notice that can be rewritten using the distributive property: Here, (complement law in Boolean algebra). So,
Thus, the expression becomes:
Step 2: Factorize from :
We can factor from : Here, (nullification law). So:
The expression now simplifies to:
Step 3: Simplify :
In Boolean algebra, is the simplest form, as no further factoring or simplification can be done. Comparing this with the right-hand side (), observe that covers the term from the original expression.
Thus, the simplified expression matches:
Conclusion:
This proves the given equivalence.
Would you like to explore another example or get additional explanations? Here are some related questions:
- What are the basic Boolean algebra laws used in such simplifications?
- How does the complement law work in practical logic circuits?
- Why does reduce to in this context?
- Can be factored further in any other scenarios?
- How would this simplify if we assigned specific Boolean values to and ?
Tip: Boolean algebra simplifications often use distributive, complement, and nullification laws. Practice combining them for efficient results!
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Math Problem Analysis
Mathematical Concepts
Boolean Algebra
Simplification
Formulas
x + x' = 1
x + 1 = 1
yz = y(x + x')
Theorems
Complement Law
Nullification Law
Distributive Property
Suitable Grade Level
Grades 11-12
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