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
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