LogicIB Mathematical Studies SL

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Introduction to symbolic logic

Mathematical logic deals with the conversion of worded statements or arguments into symbols, and how we can apply rules of deduction to them.› The philosopher GW Leibniz (1646-1716) is credited with the first

development of symbolic logic.› The basis of this logic is to take “sub-statements” and replace them with

a single (usually p, q, r) letter. For example if the statement “It is raining” is represented

by Pand the statement “It is cloudy” is represented by Q;

We can reduce the sentence “If it is raining then it is cloudy” to “If P then Q”.

Example 1

Given

P: I will do well on the test tomorrowQ: I will study tonight.

Replace P and Q for the appropriate proposition below.a) If I will study tonight then I will do well on the test tomorrow.b) I will study tonight and I will do well on the test tomorrow.c) I will not study tonight.

Solution:a) If Q then Pb) Q and Pc) Not Q

Propositions The most basic types of logical statements are called

propositions.› A proposition can be either proved true or false › A proposition itself contains no logical connectives like “or”, “and”, etc

Examples of propositions are:› The cow is green.› I went to the movie.› She will travel to Thailand› 2x + 3 < 13

These are all propositions because the truth value of the statement can be determined (even if difficult in some cases)

Statements that are opinions (ie subjective) are not considered propositions (because they cannot have their truth value determined), e.g.› I like watermelon.› That puppy is cute.

Questions are also not propositions.

A truth value indicated the

extent to which a proposition is true.In Maths true =1

and false = 0.

Example 2

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Implication When a proposition (p) being true means that another

proposition (q) must also be true, then we can say that p implies q.› Mathematically this compound statement “If p then q” can be written

and is called an implication.› The first part of an implication is called the antecendent› The second part is called the consequent.

The logic symbol for an implication is => › i.e P => Q means “If p then q”

An implication is FALSE when the antecedent is true but the consequent is false. This should be obvious from the definition of the implication. An implication is TRUE in all other cases.

When translating symbolic

statements for implications into words make sure

to include the “if” part

The FALSE case is the

only one that “breaks the promise”.

Eg1) Implication example Write down the cases when the statement:

› “When I say I am going to do something, then I do it”

is true.

Remember the only time this implication statement will be false is when I don’t follow through on the promise. Hence it is true:› When I say I am going to do something and I do it.› When I don’t say I am going to do it and I don’t do it.› When I don’t say I am going to do something and I do it.

Equivalence Two propositions P and Q are called equivalence

statements if each implies the other .› Mathematically this compound statement means “P if and only if Q”

The logic symbol for an equivalence is <=> › Therefore if P is true then Q is true.› Similarly if P is false then Q is false.

Hence an equivalence statement is FALSE when one proposition is true and the other is false or vice versa.

Eg2) Equivalence example Determine the truth value of:

› “If and only if I study for my test then I will do well”

if I studied for my test but I did not do well.

Since one of the sub-propositions of the statement is true and one is false then we know the overall statement must be false. Hence the truth value is 0.

Negation (the “not” case) A proposition and its negation have opposite truth value.

› So if a proposition P is true, then the negation of p, written as is false.

› Similarly if the proposition P is false then the negation is true.

In terms of algebraic expressions such as equations and inequalities, the negated proposition is the complement of the solution set.› e.g if has a true solution of x=3 so the negation is

¬p

2x + 3 =9 x ≠3

Eg3) Negation example Write down the negation of the following statements:

› It is sunny› 2(x+3)>6› The sun is not a star›

Solutions:› It is not sunny›

› The sun is a star› p

¬p

2 x + 3( ) ≤6

Exercises 17D

Exercises 17A.1

Exercises 17A.1 (cont)

Homework