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Logic

In logic a proposition is a statement that has the answer true (T) or false (F)

Examples:

Today is Friday

My name is Fred

It is sunny

What day is it?

This is not a proposition.

Propositions are represented by the letters p, q , r in the IB exams

Examples:

p

Today is Friday

q

It is sunny

r

I will go for a walk

** The above three propositions will be used in this section on logic

 

 

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Logic, Sets and Probability

Logical operators

Joining propositions together with logical operators forms compound propositions.

1.

Negation
negation

The words 'not' or 'do not' etc. is put in front of the original proposition.

 

Example:

 

 

 

 

p

Today is not Friday

 

 

 

 

 

NB. Negation is not the opposite of the original proposition i.e. "if the hotel is full" is the first proposition, then the negation is it's "not full".  We can't say "empty", which would be the opposite.

 

Given the proposition

 

 

q

"the hotel is full"

 

then

q

is "the hotel is not full"

 

(we can't say its empty, as it may have 1 room occupied or 2 etc.)

 

Truth tables

For each compound proposition we can calculate its truth value using a Truth Table.

p

p

T

F

F

T

If the original proposition is True then the negation of it must be False.

 

2.

Conjunction
and

The word AND is used to join two propositions together.

 

 

pq

Today is Friday and it is sunny.

 

p

q

pq

T

T

T

T

F

F

F

T

F

F

F

F

 

 

 

With conjunction the overall proposition is only True when both of the propositions are true.

 

3.

Disjunction
disjunction

The word OR is used to join the two propositions together

 

 

pvq

Today is Friday or it is sunny (or both)

 

p

q

pvq

T

T

T

T

F

T

F

T

T

F

F

F

 

 

 

 

Only one of the propositions has to be True to make the compound proposition true.

 

4.

Exclusive disjunction
vunderscore

The word OR is again used to join the two propositions together, but this time it is not both.

 

 

pvuq

Today is Friday or it is sunny, but not both.

 

p

q

pvuq

T

T

F

T

F

T

F

T

T

F

F

F

 

 

 

Only one of the original propositions must be true for the compound proposition to be true.

 

5

Implication
Implication

The words If ..... Then ..... are used to join the two propositions together.

(The first proposition implies the second) Often thought of as the first proposition being the hypothesis and the second the conclusion.

 

 

implication

If it is Friday then it is sunny

 

p

q

implication

T

T

T

T

F

F

F

T

T

F

F

T

 

 

 

The compound proposition is only false when the first proposition is True and the second False.

 

6.

Equivalence
Equivalence

The words If and only if are used to join the two propositions.

 

 

equivalence

It is Friday if and only if it is sunny.

 

p

q

equivalence

T

T

T

T

F

F

F

T

F

F

F

T

 

 

 

The compound proposition is True if the truth values of the two propositions are the same (equivalent)

 

7

Converse, Inverse and Contrapositive

 

 

Given p Implication q : If it is Friday then it is sunny, is the implication

 

 

Then the Converse is q Implication p : If it is sunny then it is Friday.

 

 

The Inverse is negation p Implication negation q : If it is not Friday then it is not sunny.

 

 

The contrapositive is negation q Implication negation p : If it is not sunny then it is not Friday.

Two pairs of the above statements are logically equivalent (have the same truth values as can be seen from the truth table below)

truth table

Implication and the contrapositive are logically equivalent as are the Converse and inverse.

 

8

Tautology and Contradiction

 

A tautology is a compound proposition (statement) that always has the truth value True, whatever the truth values of the individual propositions. A contradiction always has the truth value False.

 

Example

tautology is a contradiction as can be seen from the truth table below.

TT2

 

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