2. Introduction to set operations II

In the Venn diagram below, the area in the yellow colour represents $B - A$.

 

The symmetric difference of two sets $A$ and $B$ is the set of union elements which belong to $A$ but not to $B$, and symbolically we write $A – B$ and read as "$A$ minus $B$". If it belong to $B$ but not to $A$, symbolically we write $B – A$ and read as "$B$ minus $A$". Symbolically we write $(A–B)∪(B–A)$. It is denoted by $A Δ B$.

 

Symbolically we write $A\Delta B = \{x:x\in A-B\ \mathrm{or}\ x\in B-A\}$.

 

 

In the Venn diagram below, the area in the blue colour represents $A Δ B$.

 

Let $U$ be the universal set, and $A$ be a subset of $U$. Then the complement of $A$ is the set of all elements of $U$ which are not the elements of $A$. Symbolically, we write $A′$ to denote the complement of $A$ with respect to $U$. 

 

Symbolically we write $A^\prime = \{x: x\in U\ \mathrm{and}\ x\notin A\}$. Obviously $A^\prime=U-A$.

 

Note: The complement of a set $A$ can be looked upon, alternatively, as the difference between a universal set $U$ and the set $A$.

 

In the Venn diagram below, the area in the blue colour represents $A'$.