72:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The chain of number sets?"}]]}] a9:T75f,

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

73:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$a9"}}]}]]}] 74:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Empty set?"}]]}] 75:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["The set with no element: ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\varnothing

"}}],"."]}]]}] 76:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Is ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\{\\varnothing\\}

"}}]," empty?"]}]]}] 77:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["No. It has one element, ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\varnothing

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\\{a, b\\}

"}}]," and ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\{c, d\\}

"}}],"."]}]]}] 7e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Subset?"}]]}] 7f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B \\subseteq A

"}}],": every element of ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B

"}}]," is in ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}],"."]}]]}] 80:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Proper subset?"}]]}] aa:T48b,

B \ne A

81:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B \\subset A

"}}],": ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B \\subseteq A

"}}]," and ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$aa"}}],"."]}]]}] 82:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Which set is a subset of every set?"}]]}] 83:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["The empty set ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\varnothing

"}}],"."]}]]}] 84:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Power set?"}]]}] 85:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

P(A)

"}}],": the set of all subsets of ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}],"."]}]]}] 86:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Number of subsets of an ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

n

"}}],"-element set?"]}]]}] 87:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

2^n

"}}]}]]}] 88:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Number of proper subsets? Non-empty proper subsets?"}]]}] ab:T44c,

2^n - 1

ac:T44c,

2^n - 2

89:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ab"}}],"; ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ac"}}],"."]}]]}] 8a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Universal set?"}]]}] 8b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["The set ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

U

"}}]," containing all sets of the discussion; the rectangle in a Venn diagram."]}]]}] 8c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Square vs round bracket in an interval?"}]]}] 8d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Square: end point included. Round: excluded."}]]}] ad:T743,

\{x \in \mathbb{R} : -2 \le x < 3\}

8e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Interval for ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ad"}}],"?"]}]]}] 8f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

[-2, 3)

"}}]}]]}] 90:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Bracket next to ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\infty

"}}],"?"]}]]}] 91:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Always round."}]]}] 92:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Union?"}]]}] 93:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A \\cup B

"}}],": in ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}]," or in ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B

"}}]," (or both)."]}]]}] 94:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Intersection? Disjoint sets?"}]]}] ae:T43c,

A \cap B = \varnothing

95:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A \\cap B

"}}],": in both. Disjoint: ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ae"}}],"."]}]]}] 96:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Difference ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A - B

"}}],"?"]}]]}] af:T736,

A - B \ne B - A

97:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["In ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

A

"}}]," but not in ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

B

"}}],". ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$af"}}]," in general."]}]]}] 98:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Complement?"}]]}] b0:T5f6,

A' = U - A

99:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b0"}}],"."]}]]}] b1:T499,

A \cup A'

b2:T499,

A \cap A'

9a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b1"}}]," and ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b2"}}],"?"]}]]}] 9b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

U

"}}]," and ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\varnothing

"}}],"."]}]]}] 9c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"De Morgan's laws?"}]]}] b3:Tb84,

(A \cup B)' = A' \cap B'

b4:Tb84,

(A \cap B)' = A' \cup B'

9d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b3"}}],"; ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b4"}}],"."]}]]}] 9e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Distributive law?"}]]}] b5:Tae1,

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

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\\cup

"}}],", ","$Lb6"," swapped."]}]]}] a0:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"How to prove a set identity?"}]]}] a1:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Show each side is a subset of the other, using a general element."}]]}] a2:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Union formula for two sets?"}]]}] b7:Tabd,

n(A \cup B) = n(A) + n(B) - n(A \cap B)

a3:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b7"}}]}]]}] a4:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Union formula for three sets?"}]]}] a5:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Singles ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

-

"}}]," pairs ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

+

"}}]," triple intersection."]}]]}] a6:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Which topics are formative only or deleted here?"}]]}] a7:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Practical counting problems are not board assessed; Mathematical Induction is formative only; Mathematical Reasoning is deleted."}]]}]

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