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Truth Tables and Venn Diagrams

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Page 1: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Truth Tables and Venn Diagrams

Page 2: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Truth Tables

Truth Tables are a very convenient way to organize the truth values of statements.

Disjunction

p q

Conjunction

p q p q p q

Page 3: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Truth Tables

p q ~p p q

Page 4: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Truth Tables

p q ~p p q

Page 5: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Venn diagrams:• show relationships between different sets of data.

• can represent conditional statements.

• is usually drawn as a circle.

– Every point IN the circle belongs to that set.

– Every point OUT of the circle does not.

Example: A = poodle ... a dog

B = horse ... NOT a dog

.BDOGS

.A

Page 6: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

For all..., every..., if...then...

All right angles are congruent.

Congruent Angles

Right Angles

Example1:

Example 2: Every rose is a flower.

Flower

Rose

Example 3: If two lines are parallel, then they do not intersect.

lines that do not intersect

parallel lines

Page 7: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

To Show Relationships using Venn Diagrams:

Blue or Red (includes Purple) … AB

A B

A B

Page 8: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Example:

Twenty-four members of Mu Alpha Theta went to a Mathematics conference.

One-third of the members ran cross country.

One sixth of the members were on the football team .

Three members were on cross country and football teams.

The rest of the members were in the band.

How many were in the band?

Hint: Draw a Venn Diagram and take one sentence at a time...

Page 9: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Solution:

• Three members were on cross country and football teams…

3The above sentence tells you two draw overlapping circles and put 3 in CCF

Twenty-four members of Mu Alpha Theta went to a Mathematics conference.

One-third of the members ran cross country.24 / 3 = 8; 8 members run cross country. So put 5 in cross country as there are already 3 in cross country.

CC Football

5

One sixth of the members were on the football team .

24/6 = 4; 4 members play football. So put 1 in football as there are already 3 in football.

1

Continued….

Page 10: Truth Tables and Venn Diagrams. Truth Tables Truth Tables are a very convenient way to organize the truth values of statements. Disjunction pq Conjunction

Example: Continued……

• The rest of the members were in the band. How many were in the band?

• Out of 24 members in Mu Alpha Theta, 9 play football or run cross country.

• Therefore, 15 are in the band.

CC Football

35 1

Band

15

Mu Alpha Theta