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What does the following expression evaluate to if a = true and b = false?
`!(a && b)`
true
What does the following expression evaluate to if a = true and b = false?
`a || b && !a`
true
What does the following expression evaluate to if a = false and b = true?
`!a && (b || a)`
true
What does the following expression evaluate to if a = false and b = false?
`!(a || b)`
true
What does the following expression evaluate to if a = true and b = true?
`a && b`
true
What does the following expression evaluate to if a = true and b = false?
`a || b`
true
What does the following expression evaluate to if a = false?
`!a`
true
What does the following expression evaluate to if a = true?
`!a`
false
What does the following expression evaluate to if a = true and b = false and c = true?
`(a && b) || (!a && c)`
false
What does the following expression evaluate to if a = true and b = false and c = true?
`!(a && b) && c`
true
Explain Proof by Simplification.
Showing boolean expression equivalence by using boolean properties and identities to transform one expression into another.
Explain Proof by Testing All Cases.
Showing boolean expression equivalence by using truth tables to demonstrate that both expressions produce the same output for all possible inputs.
Why are DeMorgan's Theorems important?
They are crucial for simplifying and negating complex boolean expressions, especially when dealing with AND and OR operators.
What is the significance of the Distributive Law in boolean algebra?
It allows you to expand or factor boolean expressions, which is useful for simplification and manipulation.
What is the order of operations in boolean expressions?
NOT (!), then AND (&&), then OR (||).
How do Commutative and Associative Laws simplify boolean expressions?
They allow rearranging and regrouping terms without changing the expression's value, making it easier to simplify.
Explain the Idempotent Law.
Applying the same input to AND or OR operations results in that same input (a && a == a, a || a == a).
Why are truth tables useful for proving boolean equivalence?
They provide a systematic way to check all possible input combinations and verify that two expressions produce the same output.
Describe the Consensus Theorem.
The Consensus Theorem helps simplify expressions with specific patterns, such as a || (!a && b) == a || b.
How does double negation simplify a boolean expression?
Double negation cancels out, i.e., !(!a) == a.
What is Boolean Algebra?
A branch of algebra in which the values of variables are either true or false, usually denoted as 1 or 0.
What is a Boolean Expression?
An expression that evaluates to either true or false.
What is the AND operator?
A logical operator that returns true if both operands are true, and false otherwise.
What is the OR operator?
A logical operator that returns true if at least one of the operands is true, and false otherwise.
What is the NOT operator?
A logical operator that negates the operand; if the operand is true, it returns false, and vice versa.
What is a Truth Table?
A table that shows all possible input combinations and their corresponding outputs for a boolean expression.
What is the Commutative Law?
A law stating that the order of operands does not affect the result for AND and OR operations (a && b == b && a, a || b == b || a).
What is the Associative Law?
A law stating that the grouping of operands does not affect the result for AND and OR operations (a && (b && c) == (a && b) && c, a || (b || c) == (a || b) || c).
What is the Distributive Law?
A law that allows distributing AND over OR: a && (b || c) == (a && b) || (a && c).
What are DeMorgan's Theorems?
Theorems that describe how to negate complex expressions: !(a && b) == !a || !b, !(a || b) == !a && !b.