Boolean Algebra Properties Quiz

Commutative Properties

• States that the order of variables does not affect the outcome.
• For disjunction: ( x \lor y = y \lor x )
• For conjunction: ( x \land y = y \land x )

Associative Properties

• Focuses on how variables are grouped in expressions.
• For conjunction: ( (x \land y) \land z = x \land (y \land z) )
• For disjunction: ( (x \lor y) \lor z = x \lor (y \lor z) )

Identity Elements

• Defines specific values that do not change the variable.
• For conjunction: ( x \land \text{TRUE} = x )
• For disjunction: ( x \lor \text{FALSE} = x )

Double Negative

• Shows that negating a negation returns the original value.
• Expressed as: ( \neg(\neg x) = x )

Idempotence

• Indicates that repeating an operation does not change the outcome.
• For conjunction: ( x \land x = x )
• For disjunction: ( x \lor x = x )

Distributive Properties

• Describes how conjunction distributes over disjunction and vice versa.
• Conjunction over disjunction: ( x \land (y \lor z) = (x \land y) \lor (x \land z) )
• Disjunction over conjunction: ( x \lor (y \land z) = (x \lor y) \land (x \lor z) )

Complementation

• Discusses the relationship between a variable and its negation.
• A variable and its negation yield FALSE when combined with conjunction: ( x \land (\neg x) = \text{FALSE} )
• A variable and its negation yield TRUE when combined with disjunction: ( x \lor (\neg x) = \text{TRUE} )

De Morgan's Laws

• Provides transformation rules for negating conjunctions and disjunctions.
• The negation of a conjunction: ( \neg(x \land y) = (\neg x) \lor (\neg y) )
• The negation of a disjunction: ( \neg(x \lor y) = (\neg x) \land (\neg y) )

Annihilation

• Reflects the effect of specific constant values on variables.
• Combining a variable with FALSE through conjunction results in FALSE: ( x \land \text{FALSE} = \text{FALSE} )
• Combining a variable with TRUE through disjunction results in TRUE: ( x \lor \text{TRUE} = \text{TRUE} )

Implication (x → y)

• Represents a logical relationship where one condition leads to another.
• Expressed as: "if x, then y"
• False only when ( x = \text{TRUE} ) and ( y = \text{FALSE} ); otherwise, it is TRUE.

Biconditional (x ⟺ y)

• Indicates mutual conditions between two variables.
• Means "x if and only if y"
• True only when both x and y share the same truth value; otherwise, it is FALSE.