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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
What are the general steps to simplify a boolean expression?
1. Apply boolean laws. 2. Use DeMorgan's Theorems. 3. Apply Distributive Law. 4. Simplify using Commutative and Associative Laws. 5. Repeat until simplified.
What are the steps to create a truth table for a boolean expression?
1. List all input variables. 2. Create columns for all possible input combinations. 3. Create columns for intermediate expressions. 4. Create a final column for the expression's output.
What are the steps to prove equivalence of boolean expressions using simplification?
1. Choose one expression. 2. Apply boolean laws and theorems. 3. Simplify the expression step-by-step. 4. Check if the simplified expression matches the other expression.
What are the steps to prove equivalence of boolean expressions using truth tables?
1. Create a truth table for each expression. 2. Compare the output columns of both truth tables. 3. If the output columns are identical, the expressions are equivalent.
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.