Complement laws
In mathematics, the complement laws refer to two fundamental properties related to sets and their complements
In mathematics, the complement laws refer to two fundamental properties related to sets and their complements.
1. Law of Complementarity: The law of complementarity states that the intersection of a set A and its complement (A’) is the empty set (∅). In other words, any element that belongs to set A cannot belong to its complement, and vice versa. The intersection of A and A’ is denoted as A ∩ A’ = ∅.
For example, let’s consider a universal set U that contains all integers from 1 to 10. If we define set A as {2, 4, 6, 8, 10}, its complement A’ would be {1, 3, 5, 7, 9}. The intersection of A and A’ is empty because they do not share any common elements.
2. Law of Double Complement: The law of double complement states that the complement of a complement of a set is the set itself. In other words, taking the complement of a set twice will yield the original set. This law is often represented as (A’)’ = A.
Continuing with the previous example, if we take the complement of A’ (the complement of set A), we would obtain set A again. The complement of A’ is {2, 4, 6, 8, 10}, which is equal to A.
These complement laws are essential for understanding set theory and set operations. They help establish relationships between sets and their complements, allowing us to analyze and manipulate sets in various mathematical contexts.
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