Contemporary Mathematics

# Key Concepts

### 1.1Basic Set Concepts

• Identify a set as being a well-defined collection of objects and differentiate between collections that are not well-defined and collections that are sets.
• Represent sets using both the roster or listing method and set builder notation which includes a description of the members of a set.
• In set theory, the following symbols are universally used:
ℕ - The set of natural numbers, which is the set of all positive counting numbers.
$ℕ={1,2,3,...}ℕ={1,2,3,...}$

ℤ - The set of integers, which is the set of all the positive and negative counting numbers and the number zero.
$ℤ={...,−2,−1,0,1,2,...}ℤ={...,−2,−1,0,1,2,...}$

ℚ - The set of rational numbers or fractions.
$ℚ={ pq|pandqare integers andq≠0 }ℚ={ pq|pandqare integers andq≠0 }$
• Distinguish between finite sets, infinite sets, and the empty set to determine the size or cardinality of a set.
• Distinguish between equal sets which have exactly the same members and equivalent sets that may have different members but must have the same cardinality or size.

### 1.2Subsets

• Every member of a subset of a set is also a member of the set containing it. $A⊆BA⊆B$
• A proper subset of a set does not contain all the members of the set containing it. There is a least one member of set $BB$ that is not a member of set $AA$. $A⊂BA⊂B$
• The number subsets of a finite set $AA$ with $n(A)n(A)$ members is equal to 2 raised to the $n(A)n(A)$ power.
• The empty set is a subset of every set and must be included when listing all the subsets of a set.
• Understand how to create and distinguish between equivalent subsets of finite and infinite sets that are not equal to the original set.

### 1.3Understanding Venn Diagrams

• A Venn diagram is a graphical representation of the relationship between sets.
• In a Venn diagram, the universal set, $UU$ is the largest set under consideration and is drawn as a rectangle. All subsets of the universal set are drawn as circles within this rectangle.
• The complement of set $AA$ includes all the members of the universal set that are not in set $AA$. A set and its complement are disjoint sets, they do not share any elements in common.
• To find the complement of set $AA$ remove all the elements of set $AA$ from the universal set $UU$, the set that includes only the remaining elements is the complement of set $AA$, $A′A′$.
• Determine the complement of a set using Venn diagrams, the roster method and set builder notation.

### 1.4Set Operations with Two Sets

• The intersection of two sets, $A∩BA∩B$ is the set of all elements that they have in common. Any member of $AA$ intersection $BB$ must be is both set $AA$ and set $BB$.
• The union of two sets, $A∪BA∪B$, is the collection of all members that are in either in set $AA$, set $BB$ or both sets $AA$ and $BB$ combined.
• Two sets that share at least one element in common, so that they are not disjoint are represented in a Venn Diagram using two circles that overlap.
• The region of the overlap is the set $AA$ intersection $BB$, $A∩B.A∩B.$
• The regions that include everything in the circle representing set $AA$ or the circle representing set $BB$ or their overlap is the set $AA$ union $BB$, $A∪B.A∪B.$
• Apply knowledge of set union and intersection to determine cardinality and membership using Venn Diagrams, the roster method and set builder notation.

### 1.5Set Operations with Three Sets

• A Venn diagram with two overlapping sets breaks the universal set up into four distinct regions. When a third overlapping set is added the Venn diagram is broken up into eight distinct regions.
• Analyze, interpret, and create Venn diagrams involving three overlapping sets.
• Including the blood factors: A, B and Rh
• To find unions and intersections.
• To find cardinality of both unions and intersections.
• When performing set operations with three or more sets, the order of operations is inner most parentheses first, then fine the complement of any sets, then perform any union or intersection operations that remain.
• To prove set equality using Venn diagrams the strategy is to draw a Venn diagram to represent each side of the equality or equation, then look at the resulting diagrams to see if the regions under consideration are identical. If they regions are identical the equation represents a true statement, otherwise it is not true.
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