Set-Notation-And-Venn-Diagrams Aqa Higher

Set notation

What is a set?

• A set is a collection of elements

  • Elements could be anything

    • Numbers, letters, coordinates, …

• You could describe a set by writing its elements inside curly brackets {}

  • {1, 2, 3, 6} , is the set of factors of 6

• If the set of elements follow a rule then you can write this using a colon inside the curly brackets {… : …}

  • The bit before the colon is the type of element

  • The bit after the colon is the rule 

    • {x is a positive integer : x² < 30} is the set of positive integers which, when squared, are less than 30

    • This is equal to {1, 2, 3, 4, 5}

  • The colon is often read as ‘such that’ 

  • If no type is specified, x can take any value (fractions, decimals, irrationals, …)

    • {x: x² < 30} means any value whose square is less than 30

  • { (x, y) : y = mx + c } would mean the coordinates (x, y) where y = mx + c

    • I.e. The set of all possible coordinates that lie on the line y = mx + c

  • A colon can also be replaced by a vertical bar

    • {x | x² < 30}

What do I need to know about set notation?

• is the universal set (the set of everything)

  • For example, if we are only interested in factors of 24 then = {1, 2, 3, 4, 6, 8, 12, 24}

  • You may see alternative notations used for  

    • U is a common alternative (different to for union!)

    • S or the Greek letter ξ (xi) may also be seen

• We use upper case letters to represent sets (A, B, C, …) and lower case letters to represent elements (a, b, c, …)

• means “is an element of“

  • E.g. <img alt=”3 element of open curly brackets 1 comma 2 comma 3 close curly brackets” data-mathml=”<math ><semantics><mrow><mn>3</mn><mo>&#8712;</mo><mfenced open=”{” close=”}”><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></mfenced></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation></semantics></math>” height=”22″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2222%22%20width%3D%2281%22%20wrs%3Abaseline%3D%2216%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E3%3C%2Fmn%3E%3Cmo%3E%26%23×2208%3B%3C%2Fmo%3E%3Cmfenced%20close%3D%22%7D%22%20open%3D%22%7B%22%3E%3Cmrow%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmn%3E3%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmfenced%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math1b02d2e8b97d8ef2690c5458142’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUN