Sets

Sets refers to the group of elements or a collection of elements

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Sets fundamental Concepts

From the defination of a set we can draw out two ways of representing sets namely vasual and list notation

Sets can be finite or infinite

A = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10}

B = {1, 3, 4, 5, 6, 7, ...}

Set A represent a finite kind of a set stating that it has a specific start and end it indicates that set A starts with number 1 (one) and ends with number 10 (ten) from a given example

Set B is an example of an infinite set meaning that it has no end, it has started with number 1 (one) but does not have an end from the above example

Repeated elements are listed once

A = { z, n, b, a, z, z, n, a } = {a, b, n, z }

no repetitive elements should be listed more than once

There is no order in a set

{7, 2, 1} = {1, 2, 7 }

The order of a sets really does not impact the change of a given sets

Common sets

Natural numbers

below are examples of natural number

{0, 1, 2, 3, 4, ...} or {1, 2, 3, 4, ...}

Integers

below are examples of Integers these are numbers that are positive or negatives

{..., -3, -2, -1, 0, 1, 2, 3, ...}

Rational numbers

below are examples of Rational number these are fractional numbers

$\{...,\frac{1}{1},\frac{1}{2},\frac{1}{3},\; ...\}$

D = {white, black, grey }

White is an element of D , white $\in$ D

Blue is not an element of D, Blue $\notin$ D

Cardinality of a set

The Cardinality of D is 3 (Size), |D| = 3

Cardinality refers to the number of elements which makes up a set

Empty set , $\varnothing$ = { }

Intersection sets, Union of a set and Venn diagram

Find the Intersection of the given sets

A = {3, 7, 9, 10, 12, 17}

B = {2, 3, 9, 11, 12, 18, 20 }

To find the Intersection of set A and B we have to find out the elements which are common in both sets

A $\cap$ B = {3, 9, 12}

Find the Intersection of the given sets

M = {a, b, c, d, e }

N = {f, g, h }

M $\cap$ N = { } or $\varnothing$

Find the Union of the given sets

p = {1, 4, 5, 7, 9 }

Q = {4, 9, 10, 13 }

Union means combining these sets to a joint set

P $\cup$ Q = {1, 4, 5, 7, 9, 10, 13}

How to solve a venn diagram with examples

The diagram below shows three sets A, B and C. Given that (A∪B∪C)=50 find

(i).   the value of x

(ii).   $\mathrm{n(AUB)}$

(iii).   ${\mathrm{n(BUC)}}^{\mathrm{,}}$

(iv).   $n(A^{\mathrm{,}}n{C}^{\mathrm{,}})$

Solution:

(i).   (A∪B∪C)=50

(A∪B∪C)=50. This equation reads that the elements in A union B union C adds upto $50$

$\mathrm{10\; +\; 3x\; +\; 3\; +\; x\; +\; 5\; =\; 50}$

$\mathrm{10\; +\; 5\; +\; 3\; 3x\; +\; x\; =\; 50}$

$\mathrm{18\; +\; 3x\; +\; x\; =\; 50}$

$\mathrm{3x\; +\; x\; =\; 50\; -\; 18}$

$\mathrm{4x\; =\; 50\; -\; 18}$

$\mathrm{4x\; =\; 32}$

$\frac{x}{4}=\frac{\mathrm{32}}{4}$

Answer: $\mathrm{x\; =\; 8}$

(ii).   $\mathrm{n(AUB)}$ reads numbers in A union B means elements in A plus elements in set B

n(AUB) = 10 + 3x + 3 + x

Note that X = 8

n(AUB) = $\mathrm{10\; +\; 3(8)\; +\; 3\; +\; 8}$

n(AUB) = $\mathrm{10\; +\; 24\; +\; 3\; +\; 8}$

Answer:   n(AUB) = $45$

(iii).  ${\mathrm{n(BUC)}}^{\mathrm{,}}$

${\mathrm{n(BUC)}}^{\mathrm{,}}$ reads number of element B Union C complement, meaning elements that are not in union B and C in this case its 10 and 7

${\mathrm{n(BUC)}}^{\mathrm{,}}=\mathrm{10\; +\; 7}$

Answer:   ${\mathrm{n(BUC)}}^{\mathrm{,}}=17$

(iv).   $n(A^{\mathrm{,}}n{C}^{\mathrm{,}})$ reads A complement Intersection C complement, meaning number of elements that are not in both sets. in this case it is 3 and 7

$n(A^{\mathrm{,}}n{C}^{\mathrm{,}})=\mathrm{3\; +\; 7}$

Answer:  $n(A^{\mathrm{,}}n{C}^{\mathrm{,}})=10$

Venn diagram Example

Draw a venn diagram from the following sets by representing the Intersection and Union of sets

X = {6, 7, 8, 9, 10}

Y = {5, 7, 8, 11, 12}

X $\cap$ Y = {7, 8}

X $\cup$ Y = {5, 6, 7, 8, 9, 10, 11, 12}

The venn diagram

Derive answers from a venn diagram

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ECZ Mathematics sets pdf