Solving Venn Diagrams.ppt - MathXTC

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```							           SOLVING
VENN DIAGRAMS

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The universal Set, U, and the sets A,
B and Contained within it are such
that:
n(U )  74     n( A  B)  10
n( A)  25     n( B  C )  11
n( B )  34    n( A  C )  16
n(C )  37     n( A  B  C )  8

Find: n( A  B  C )
SOLVING
VENN DIAGRAMS

Method 1:
Algebraic Approach

www.mathxtc.com
The universal Set, U, and the sets A,
B and Contained within it are such
that:
n(U )  74
A                 B
n( A  B)  10
n( A)  25       n( B  C )  11
n( B )  34      n( A  C )  16
n(C )  37       n( A  B  C )  8

Find:
n( A  B  C )                C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                      74

n(U )  74
A                 B
n( A  B)  10
n( A)  25       n( B  C )  11
n( B )  34      n( A  C )  16
n(C )  37       n( A  B  C )  8

Find:
n( A  B  C )                C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                            74

n(U )  74
A                       B
n( A  B)  10
25               34
n( A)  25       n( B  C )  11
n( B )  34      n( A  C )  16
n(C )  37       n( A  B  C )  8

Find:                              37

n( A  B  C )                C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                                    74

n( A  B)  10
A                                B

10
25                      34
n( B  C )  11
n( A  C )  16
n( A  B  C )  8            16            11

Find:                                37

n( A  B  C )                  C
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                                    74

n( A  B)  10
A                                B

10
25                      34
n( B  C )  11
n( A  C )  16
n( A  B  C )  8            16            11

Find:                                37

n( A  B  C )                  C                      8
The universal Set, U, and the sets A,
B and Contained within it are such
that:                                                    74

n( A  B)  10
A                                B

10
25                      34
n( B  C )  11
n( A  C )  16
x
n( A  B  C )  8            16            11

Find:                                37

n( A  B  C )                  C                      8
74

A                                                       B

10
25                                             34
10 - x
a                               b

x
16 - x             11 - x
16                                   11

c

37
C                                   8
a + 10 + (16 – x) = 25                                                              74

a=x–1                   A                                                      B

10
25                                            34
10-x
-
xa 1                            b

x
16-x              11-x
16                                  11

c

37
C                                   8
a + 10 + (16 – x) = 25                                                              74

a=x–1                   A                                                      B
b + 10 + (11 – x) = 34                              10
25                                            34
 b = x + 13
10-x
b
x + 13
x-1

x
16-x              11-x
16                                 11

c

37
C                                   8
a + 10 + (16 – x) = 25                                                                  74

a=x–1                         A                                                    B
b + 10 + (11 – x) = 34                                     10
25                                          34
 b = x + 13
10-x
x-1                            x+13

n( A  B  C )  7                                         x
16-x              11-x
16                                11
c

37
37 + (x – 1) + (10 – x) + (x + 13) = 74 – 8
C                                 8
59 + x = 66
x=7
a + 10 + (16 – x) = 25                                                           74

a=x–1                         A                                             B
b + 10 + (11 – x) = 34                                    10
25                                   34
 b = x + 13
3
6                    20

n( A  B  C )  7                                       7
9            4
16                         11
17

37
37 + (x – 1) + (10 – x) + (x + 13) = 74 – 8
C                           8
59 + x = 66
x=7
SOLVING
VENN DIAGRAMS

Method 2:
Approach

www.mathxtc.com
Here is the given information:
74

A                                B

10
25                      34

16            11

Find:
n( A  B  C )
37
C                      8
n( A  B  C )  10            74

A                                        B

10
25                              34

16                   11

37
C                        8
Try x = 4 ?                                                               7474
AA                                                  BB
n(U )  71
10
2525                                    3434
Needs to be 74
6
3                      17

4
12             7
1616                      1111
14

37
37
CC                               88
Now Try x = 7 ?                                                           7474
AA                                                 BB
n(U )  74
10
2525                                   3434
Bingo !!!
3
x=7                             6                     20
74 – 71 = 3 and                               7
4+3=7                                9             4
Is this a                     1616                     1111
coincidence? 
17

37
37
CC                               88
SOLVING
VENN DIAGRAMS

Method 3:
Reasoning Approach

www.mathxtc.com
Here is the given information:
74

A                                B

10
25                      34

16            11

Find:
n( A  B  C )
37
C                      8
74
25 + 34 + 37 = 96
A                                B
A + B + C = 66
10
25                      34
96 – 66 = 30 Surplus

10 + 16 + 11 = 37

37 – 30 = 7 Surplus                           7

16            11

Find:
n( A  B  C )         =7

37
C                      8

```
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