Introduction to Sets
Tutor-style math help
Introduction to Sets: what to notice and how to work it
Sets Numbers skill
Number classification is careful sorting. Start with the simplest equivalent form, then name every number family that applies.
What to notice first
Simplify first. A square root, decimal, or fraction may belong to a simpler category after you rewrite it.
Common student mistake
Do not call every decimal irrational. Terminating and repeating decimals are rational because they can be written as fractions.
Key formulas and cues
\(\mathbb{N}\subset\mathbb{W}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\)
\(\text{terminating or repeating decimal}\Rightarrow\text{ rational}\)
\(\sqrt{\text{non-perfect square}}\Rightarrow\text{ irrational}\)
A reliable path
- SimplifyEvaluate roots, fractions, or decimals when possible.
- Look for fraction formIf the number is a ratio of integers, it is rational.
- Name all setsA number can belong to more than one family.
Worked examples
Classify a root
Example: \(\sqrt{16}\)
- Simplify the root.
- The value is 4.
- 4 is whole, integer, rational, and real.
Answer: Whole, integer, rational, real
Classify a decimal
Example: 0.75
- Rewrite 0.75 as 75/100.
- Reduce to 3/4.
- It is a ratio of integers.
Answer: Rational
Try one before moving on
Try: Classify -6 using number sets.
Answer: Integer, rational, and real.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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Introduction to Sets: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. 0.333… is:
2. \(\sqrt2\) is:
3. Every integer is also:
Types of sets
- Empty set or null set: It has no element present in it.
- Finite set: It has a limited number of elements.
- Infinite set: It has an infinite number of elements.
- Equal set: Two sets that have the same members.
- Subsets: A set ‘\(A\)’ is called to be a subset of \(B\) if each element of \(A\) is also an element of \(B\).
- Universal set: A set that consists of all elements of other sets present in a Venn diagram.
Sets formulas
- \(\color{blue}{n\left(A\:U\:B\right)=\:n\left(A\right)\:+\:n\left(B\right)-\:n\left(A\:∩\:B\right)}\)
- \(\color{blue}{n\:\left(A\:∩\:B\right)=n\left(A\right)+n\left(B\right)-n\left(A\:U\:B\right)}\)
- \(\color{blue}{n\left(A\right)=n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(B\right)}\)
- \(\color{blue}{n\left(B\right)=\:n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(A\right)}\)
- \(\color{blue}{n\left(A\:-\:B\right)=n\left(A\:U\:B\right)-n\left(B\right)}\)
- \(\color{blue}{n\left(A\:-\:B\right)=\:n\left(A\right)-\:n\left(A\:∩\:B\right)}\)
For any two sets \(A\) and \(B\) that are disjoint,
- \(\color{blue}{n\left(A\:U\:B\right)=n\left(A\right)+\:n\left(B\right)}\)
- \(\color{blue}{A ∩ B = ∅}\)
- \(\color{blue}{n\left(A\:-\:B\right)=n\left(A\right)}\)
Sets – Example 1:
If set \(A=\) {\(a,b,c\)} and set \(B=\) {\(a,b,c,l,z,r\)}, find \(A ∩ B\).
Solution:
\(A∩B=\) {\(a,b,c\)}
Original price was: $109.99.$54.99Current price is: $54.99.
Original price was: $109.99.$54.99Current price is: $54.99.
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