venn diagrams and logic

Venn Diagrams and

Logic

Lesson 2-2

Venn diagrams:

• show relationships between different sets of data.

• can represent conditional statements.

A Venn diagram is usually drawn as a circle.

• Every point IN the circle belongs to that set.• Every point OUT of the circle does not.

A=poodle ... a dogB= horse ... NOT a dog

.B

DOGS

.A ...B dog

Many Venn diagrams are drawn as two overlapping circles.

group2group1

A BC

A is in A is not inGroup 1 Group 2

Many Venn diagrams are drawn as two overlapping circles

group2group1

A BC

B is in Group 1 AND Group 2

Many Venn diagrams are drawn as two overlapping circles

group2group1

A BC

C is in C is not inGroup 2 Group 1

Many Venn diagrams are drawn as two overlapping circles

group2group1

A BC

of the elements in group1 are in group2Someof the elements in group2 are in group1Some

Sometimes the circles do not overlap in a Venn diagram.

D is in D is not inGroup 3 Group 4

group4

group3

DE

Sometimes the circles do not overlap in a Venn diagram.

E is in E is not inGroup 4 Group 3

group4

group3

DE

Sometimes the circles do not overlap in a Venn diagram.

group4

group3

DE

of the elements in group3 are in group4Noneof the elements in group4 are in group3None

In Venn diagrams it is possible to have one circle inside another.

group5

group6

F G

F is in Group 5 AND is in Group 6

In Venn diagrams it is possible to have one circle inside another.

group5

group6

F G

G is in G is not inGroup 6 Group 5

In Venn diagrams it is possible to have one circle inside another.

of the elements in group5 are in group6Allof the elements in group6 are in group5Some

group5

group6

F G

congruent angles

All right angles are congruent.

right angles

If two angles are right angles, then they are congruent.

flower

rose

Every rose is a flower.

If you have a rose, then you have a flower.

Lines that do not intersect

parallel lines

If two lines are parallel,

then they do not intersect.

Let’s see how this works!Suppose you are given ...

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?

Use a Venn Diagram and take one sentence at a time...

• Three members were on cross country and football teams…

• Tells you two draw overlapping circles

• Put 3 marks in CCF

Use a Venn Diagram and take one sentence at a time...

• One-third of the members ran cross country.

• put 8 marks in the CC circle since there are 24 members

• already 3 marks

• so put 5 marks

in the red partIII

FootballCC

Use a Venn Diagram and take one sentence at a time...

• One sixth of the members were on the football team .

• put 4 marks in the Football circle since there are 24 members

• already 3 marks

• so put 1 mark

in the purple partIII

FootballCC

IIIII

Use a Venn Diagram and take one sentence at a time...

• 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

15 members

are in band

Mu Alpha Theta

IIIIIIIII

BandFootball

CC

Drawing and Supporting Conclusions

Law of Detachment

You are given:a true conditional statement and

the hypothesis occurs

You can conclude:that the conclusion will also occur

Law of Detachment

You are given:pq is true

p is given

You can conclude:q is true

Symbolic form

Law of Detachment

You are given: If three points are collinear, then the

points are all on one line.

E,F, and G are collinear.

You can conclude:

E,F, and G are all on one line.

Example

Law of SyllogismYou are given:Two true conditional statements and

the conclusion of the first is the hypothesis of the second.

You can conclude:that if the hypothesis of the first occurs,

then the conclusion of the second will also occur

Law of Syllogism

You are given:pq and qr

You can conclude:pr

Symbolic form

Law of SyllogismYou are given:If it rains today, then we will not

have a picnic.

If we do not have a picnic, then we will not see our friends.

You can conclude:If it rains today, then we will not see

our friends.

Example

Series of Conditionals

The law of syllogism can be applied to a

series of statements. Simply reorder

statements.

You may need to use contrapositives,

since they are logical equivalents to the original statement.

This means1) ~st is the same as2) s r is the same as3) r ~q is the same as

~t s

~r ~s

q ~r

EXAMPLE: if p q, s r, ~s t,

and r ~q; then p ______?

You might need the contrapositives:~t s or ~r ~s or q ~r

Start with p and use the law of syllogism to find the conclusion:

p q … q ~r … ~r ~s … ~s t

p t