All Flashcards
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.
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?
- 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?
- 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?
- 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?
- 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.