Laws of the Algebra of Sets The fundamental laws of the algebra of sets
Sets and logic The "and", the conjunction or the logical product The "or", the disjunction or the logical sum Logical negation
The Fundamental Laws of the Algebra of Sets
 associative law:        (A U B) U C =  A U (B U C)               and    (A ∩ B) ∩ C = A ∩ (B ∩ C ) commutative law:      A U B = B U A                                   and    A ∩ B = B ∩ A distributive law:         A U (B ∩ C) = (A U B) ∩ (A U C)       and     A ∩ (B U C) = (A ∩ B) U (A ∩ C) De Morgan's laws:     (A U B)' = A' ∩ B'                                and     (A ∩ B)' = A' U B'
The distributive law A U ( B C ) = ( A U B ) ( A U C ) shown by Venn Diagrams: Sets and Logic
In logic a statement is a sentence that is either true or false, but not both. The truth or falsity of a statement is called its truth value. Truth values can be represented as binary numbers, where 0 denotes false and 1 denotes true.
Compound statements or proposition are two or more simple statements joined by connectives. Two connectives used to make compound statements are the words “and” and “or.”
The compound statement formed by the word “and” is called the conjunction or the logical product, and denoted as p Λ q. A conjunctive statement is true only if its statements (components) are both true.
The compound statement formed by the word “or” is called the disjunction or the logical sum, and denoted as p V q. A disjunctive statement is true if either or both of its statements is true, and false only when both are false.
The truth tables for the compound statements, p Λ q and p V q, and corresponding Venn Diagrams for the intersection and the union are shown below.
 p q p Λ q 1 0 0 0 2 1 0 0 3 0 1 0 4 1 1 1 p q p V q 1 0 0 0 2 1 0 1 3 0 1 1 4 1 1 1 Observe coincidence between the four combinations in the truth tables and the corresponding four numbered areas in the Venn Diagrams, so
- first combination, that describe the case when an element doesn't belong to the set A nor B, correspond to the part of the universal set excluding areas of A and B, numbered 1.
- second combination, that describe the case when an element belong to the set A but not to B, correspond to the part of the set A which is not in B, numbered 2.
- third combination, when an element belong to B but not to A, correspond to the area that is numbered 3.
- forth combination, when an element belong to A and B, correspond to the area that is numbered 4 and which is shared region between the sets A and B.
Logical negation "not p" or ~p denotes negation of an statement p. Thus the truth value of the negation of any statement is always opposite of the truth value of the given statement, as is shown in the truth table.
 p ~p 1 0 1 2 1 0    Beginning Algebra Contents 