72:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Reflexive relation?"}]]}] ab:T404,

(a, a) \in R

73:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ab"}}]," for ",["$","strong","strong-0",{"children":"every"}]," ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

a \\in A

"}}],"."]}]]}] 74:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Symmetric relation?"}]]}] ac:T7b2,

(a, b) \in R \Rightarrow (b, a) \in R

75:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ac"}}],"."]}]]}] 76:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Transitive relation?"}]]}] ad:T96f,

(a, b), (b, c) \in R \Rightarrow (a, c) \in R

77:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ad"}}],"."]}]]}] 78:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Equivalence relation?"}]]}] 79:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Reflexive, symmetric and transitive."}]]}] 7a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Properties of the empty relation on a non-empty set?"}]]}] 7b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Symmetric and transitive, not reflexive."}]]}] 7c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Is the universal relation an equivalence relation?"}]]}] 7d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Yes."}]]}] 7e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"\"Is perpendicular to\" on lines?"}]]}] 7f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Symmetric only."}]]}] 80:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["\"Divides\" on ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\mathbb{N}

"}}],"?"]}]]}] 81:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Reflexive and transitive, not symmetric."}]]}] 82:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Two equivalence classes ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

[a]

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

[b]

"}}]," are always…?"]}]]}] 83:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Equal or disjoint."}]]}] 84:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Quick way to spot an equivalence relation?"}]]}] 85:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"\"Same value of some property\" (same remainder, same room, same year)."}]]}] 86:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Classes of \"",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

7

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

a - b

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

\\mathbb{Z}

"}}],"?"]}]]}] 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":"

7

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

0

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

6

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

n

"}}],"-element set?"]}]]}] ae:T539,

2^{n^2 - n}

89:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ae"}}]}]]}] 8a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Number of equivalence relations on ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\{1, 2, 3\\}

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

5

"}}]," (one per way of splitting into classes)."]}]]}] 8c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["How do you show a relation is ",["$","strong","strong-0",{"children":"not"}]," transitive?"]}]]}] af:T5c1,

(a, b), (b, c) \in R

b0:T5ae,

(a, c) \notin R

8d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Give pairs ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$af"}}]," with ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b0"}}],"."]}]]}] 8e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Picture: a tick table that mirrors across the diagonal. Property?"}]]}] 8f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Symmetric."}]]}] 90:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"One-one (injective)?"}]]}] b1:Tf4a,

f(x_1) = f(x_2) \Rightarrow x_1 = x_2

91:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b1"}}],"."]}]]}] 92:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Onto (surjective)?"}]]}] 93:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Every ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

y

"}}]," in the codomain has a pre-image; range = codomain."]}]]}] 94:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Bijective?"}]]}] 95:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"One-one and onto."}]]}] 96:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Number of functions from ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

m

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

n

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

n^m

"}}]}]]}] 98:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Number of one-one functions from ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

m

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

n

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

m \\le n

"}}],")?"]}]]}] b2:T736,

n(n - 1) \cdots (n - m + 1)

99:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b2"}}]}]]}] 9a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Onto functions from ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

m

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

2

"}}],"-element set?"]}]]}] b3:T44c,

2^m - 2

9b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b3"}}]}]]}] b4:T50d,

f(x) = x^2

9c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b4"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\mathbb{R} \\to \\mathbb{R}

"}}],"?"]}]]}] 9d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Neither one-one nor onto."}]]}] b5:T50d,

f(x) = x^3

9e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b5"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\mathbb{R} \\to \\mathbb{R}

"}}],"?"]}]]}] 9f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Bijective."}]]}] b6:T4c0,

f(n) = n + 1

a0:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b6"}}],", ",["$","span","code-1",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\mathbb{N} \\to \\mathbb{N}

"}}],"?"]}]]}] a1:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["One-one, not onto (",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

1

"}}]," is missed)."]}]]}] a2:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Picture: a horizontal line meets a graph twice. Conclusion?"}]]}] a3:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The function is not one-one."}]]}] a4:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Onto proof: the step students forget?"}]]}] a5:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Checking that the pre-image lies in the domain."}]]}] a6:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Finite sets of equal size: one-one and onto?"}]]}] a7:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Each implies the other."}]]}] a8:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Which topics of this chapter are deleted for 2026–27?"}]]}] a9:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Composition of functions, invertible functions, binary operations."}]]}]

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