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Mathematical Methods, Lecture Two
2. The Logical Framework
2.1 De nition, proposition, proof and related terminology
The following proposition is typical of the ones we often encounter in mathematics:
14 is an even number.
A proposition is a mathematical statement that is either true or false. Regarding the above proposition, we are not yet in a position to decide whether it is true or false because we have not de ned what an even number is. We need to introduce a de nition. A de nition is a precise and unambiguous description of a mathe- matical term: a number k is called even only if k = 2n for some integer n. We can now provide a proof that the above proposition is true. Indeed, we have 14 = 2 7 where 7 is an integer, so 14 is an even number. The result `14 is an even number' is called a theorem. A theorem is simply a valid mathematical result and a proof is an explanation as to why this result is valid. Sometimes a theorem may be referred to as a lemma; this is a preliminary result whose main role is to help us prove a forthcoming important or general theorem. A theorem may also be referred to as a corollary; this is a result whose proof relies heavily on a preceding important or general theorem.
Example 2.1.1 The proposition `13 is an even number' is false so it is not a theorem.
2.2 Truth tables, negations and compound propositions
By its very de nition, a proposition p is either true or false. If p is true, we assign to it the truth value T. If it is false, we assign to it the truth value F. The simplest truth table is displayed below. It consists of a single column that captures the two possibilities associated with the truth value of a proposition p:
p T F
The negation of a proposition p is denoted by :p and is simply the proposition `not p'. Its truth table follows:
p :p T F F T
Example 2.2.1 Let p be the proposition 4 is an even number'. Then :p is the proposition4 is not an even number'. In this case, p is true and :p is false. On the other hand, let p be the proposition π is an even number'. Then :p is the propositionπ is not an even number'. In this case, p is false and :p is true.
A compound proposition is a proposition that is built up from simpler proposi- tions using linking words such as and, or, if-then, if-and-only-if. The rest of this subsection is concerned with some fundamental compound propositions and their truth tables.
The conjunction of two propositions p and q is denoted by p ^ q and is the propo- sition `p and q'. The conjunction p ^ q is true only if the propositions p and q are both true. Its truth table is given below:
p q p ^ q T T T T F F F T F F F F
Example 2.2.5 Let p be the proposition 16 is a perfect square' and q be the proposition64 is a perfect square'. Since both propositions p and q are true, the proposition p ) q is true.
Example 2.2.6 Let p be the proposition 13 is a perfect square' and q be the proposition64 is a perfect square'. Since p is false, the proposition p ) q is true.
Example 2.2.7 Let p be the proposition 3 is an even number' and q be the propositionπ is an even number'. Since p is false, the proposition p ) q is true.
Example 2.2.8 Let p be the proposition 5 is an even number' and q be the proposition5 is not an even number'. Again, since p is false, the proposition p ) q is true. There is no contradiction in the proposition `if 5 is an even number, then 5 is not an even number' because the premise on which it rests is false.
Example 2.2.9 Let p be the proposition 5 is not an even number' and q be the proposition5 is an even number'. Now, p is true and q is false, so the proposition p ) q is false.
Example 2.2.10 Let p be the proposition 16 is an even number' and q be the propositionthe sum of the angles of a triangle is 180 degrees'. Since both p and q are true, the proposition p ) q is true. Note that the propositions p and q seem to be logically disconnected here.
Example 2.2.11 Let p be the proposition 13 is an even number' and q be the propositionthe sum of the angles of a triangle is 32 degrees'. Since p is false, the proposition p ) q is true.
Remark 2.2.12 A theorem of the form `p implies q' is usually associated with a deductive process for inferring the conclusion q from the assumed truth of the premise p. One may wonder how the conditional proposition p ) q can be relevant in capturing this idea. After all, the truth value of the conditional p ) q is entirely determined by the truth values of its constituent propositions p and q. Whether or not we can reach q from p through a series of mathematical deductions is not relevant for establishing the truth value of the conditional p ) q. In fact, in Examples 2.2.
and 2.2.11, we cannot even imagine connecting the propositions p and q. How can the conditional p ) q be the right mathematical object to represent the process of inferring q from p? The answer to this question lies in the fact that any valid mathematical reasoning used in a proof of the form p implies q' cannot produce a false proposition q by assuming the truth of a proposition p that is actually true. This means that any such reasoning either produces a true proposition q by assuming the truth of a proposition p that is actually true or produces a true or a false proposition q by assuming the truth of a proposition p that is actually false. In each of these cases, the validity of the deductive argument and hence the validity of the resulting propositionp implies q' is indeed captured by the truth value T of the conditional p ) q.
Remark 2.2.13 The fact that a valid mathematical reasoning can never lead to a proposition of the form T implies F ' provides the logical foundations for the so- calledproof by contradiction' which we will discuss among other methods of proof in subsection 2.5.
Example 2.2.14 Referring to Example 2.2.8 and Remark 2.2.12, let us construct a valid mathematical argument that establishes the truth of the proposition if 5 is an even number, then 5 is not an even number' in agreement with the truth value T of the corresponding conditional. Note that we need to assume the truth of the premise5 is an even number' and infer the conclusion `5 is not an even number'.
Indeed, if 5 is an even number, we have that 5 = 2k for some integer k. Then, since 5 = 2(5) 5, we can write that 5 = 2(2k) 5 which implies that 5 = 4k 6 + 1. Realising that 4k 6 is a multiple of 2, we obtain the equation 5 = 2(2k 3) + 1 where 2k 3 is an integer. Thus, we deduce that 5 is not an even number. Note that this is an illustration of the case where we produce a true proposition by assuming the truth of a proposition that is actually false.
The biconditional proposition denoted by p , q is the compound proposition `p if and only if q'. Its truth table is given below:
p q p , q T T T T F F F T F F F T
Example 2.2.19 Let p be the proposition 4 is even' and q be the proposition4 + 1 is odd'. The contrapositive of the conditional proposition if 4 is even, then 4 + 1 is odd' is the conditional propositionif 4 + 1 is not odd, then 4 is not even'. Here, both p and q are true and hence their negations :p and :q are both false. The conditional proposition p ) q and its contrapositive proposition (:q) ) (:p) are both true.
A nal note on terminology: The conditional proposition p ) q' is commonly known asp is sufficient for q' or, equivalently, as q is necessary for p'. Indeed, provided that the conditional proposition p ) q is true, it must have one of the formsT ) T ', F ) T ' orF ) F '. Hence, it is sufficient for p to be true in order for q to be true. However, it is not necessary for p to be true in order for q to be true, because q can be true even if p is false. On the other hand, it is necessary for q to be true in order for p to be true because if this is not the case, then p has to be false. Also note that it is not sufficient for q to be true in order for p to be true, because it may happen that q is true and p is false.
For similar reasons, the biconditional proposition p , q is known as p is necessary and sufficient for q' or, equivalently, asq is necessary and sufficient for p'.
2.3 Logical Equivalence
Consider a proposition p and its double negation :(:p). Regardless of the truth value of the proposition p, the propositions p and :(:p) have the same truth value. This is con rmed by the truth table below:
p :p :(:p) T F T F T F
In general, two propositions A and B that are built up from a single proposition p through some logical operations such as :, ^, _, ), , are called logically equivalent if they always have the same truth value; that is, for every truth value of p, the truth value of A and the truth value of B are the same.
Remark 2.3.1 In the above example, the role of A is played by p and the role of B is played by :(:p). Note that both A and B are built up from p.
We can extend this idea and apply it to propositions A and B that are built up from two propositions p and q.
Two propositions A and B that are built up from two propositions p and q are called logically equivalent if they always have the same truth value; that is, for every pair of truth values of p and q, the truth value of A and the truth value of B are the same.
Remark 2.3.2 Looking at the truth table preceding Remark 2.2.18, we realise that the proposition A given by p ) q and its contrapositive proposition B given by (:q) ) (:p) are logically equivalent. This fact can be very useful in some proofs as we will see in subsection 2.5.
Example 2.3.3 In order to show that the biconditional proposition p , q is logically equivalent to the conjunction of the conditional propositions p ) q and q ) p we construct the following truth table and focus on its last two columns:
p q p ) q q ) p p , q (p ) q) ^ (q ) p) T T T T T T T F F T F F F T T F F F F F T T T T
Example 2.3.4 Similarly, we show that the conditional proposition p ) q is logically equivalent to the proposition (:p) _ q. The relevant truth table is given below:
p q :p p ) q (:p) _ q T T F T T T F F F F F T T T T F F T T T
Example 2.3.5 We also show that the proposition :(p _ q) is logically equivalent to the proposition (:p) ^ (:q):
there exists a real number n such that n^2 = 17, there exists an integer m such that 3 < m < 12.
An existential statement can be written in the form 9 n[P (n)]. The symbol 9 is called the existential quanti er and simply means `there exists'. The property P (n) satis ed by n is enclosed in the square brackets.
Example 2.4.2 The existential statement there exists a real number n such that n^2 = 17' can be written in the form 9 real number n[n^2 = 17]'. Similarly, the existential statement there exists an integer m such that 3 < m < 12' can be written in the form 9 integer m[3 < m < 12]'.
Remark 2.4.3 The existential statements given in Example 2.4.2 are both true. In order to prove them, we simply need to nd at least one real number n and at least one integer m with the required properties. In general, in order to prove that an existential statement is true we just need to nd an example.
A universal statement is a statement which expresses the fact that all objects of a certain kind have a particular property. Examples of universal statements are the following:
for all integers n, 2n is a even number, for all integers m, if m^2 + m is even, then m is even.
A universal statement can be written in the form 8 n[P (n)]. The symbol 8 is called the universal quanti er and simply means `for all'. The property P (n) satis ed by n is enclosed in the square brackets.
Example 2.4.4 The universal statement for all integers n, 2n is a even number' can be written in the form 8 integers n[2n is even]'. Similarly, the universal statement for all integers m, if m^2 + m is even, then m is even' can be written in the form 8 integers m[(m^2 + m is even) ) (m is even)]'.
Remark 2.4.5 The rst statement in Example 2.4.4 is true and is straightforward to prove by considering separately the case where n is odd and the case where n is
even. The second statement in Example 2.4.4 is false. In order to establish that it is false we need to nd a counterexample. That is, we need to nd an integer m such that m^2 + m is even and m is odd, thereby obtaining a conditional proposition of the form T ) F which is false. We can easily see that m = 1 provides such a counterexample. In general, in order to prove that a universal statement is false we just need to nd a counterexample.
In preparation for some methods of proof that we will discuss in subsection 2.5 and also later in the course, let us write down the negation of an existential statement as well as that of a universal statement:
The negation of the existential statement 9 n[P (n)] is the universal statement
8 n[:P (n)].
The negation of the universal statement 8 n[P (n)] is the existential statement
9 n[:P (n)].
Remark 2.4.6 Equivalently, we can say that :
9 n[P (n)]
is logically equivalent
to 8 n[:P (n)] and that :
8 n[P (n)]
is logically equivalent to 9 n[:P (n)].
Example 2.4.7 Let n be an integer. The negation of the existential statement there exists an n such that n^2 = 7' is the universal statementfor all n, n^2 ̸= 7'.
Example 2.4.8 Let n be a natural number. The negation of the universal statement for all n, n^2 n' is the existential statementthere exists an n such that n^2 < n'.
2.5 Some methods of proof
In this subsection, we collect some methods of proof that we will encounter in this module.
A direct proof is the most straightforward method of proof where the truth or falsity of a given mathematical statement is established directly.
Example 2.5.3 Let us prove the existential statement `there exists an integer n such 2n^ = n^2 '. An example suffices: n = 2 is such an integer.
Example 2.5.4 Let us disprove the universal statement `for all integers n, 2n^ ̸= n^2 '.
A counterexample suffices: n = 2. Note that disproving this statement amounts to proving its negation, which is logically equivalent to the existential statement encountered in Example 2.5.3. Our counterexample' here is called anexample' there.
A proof by contradiction is based on the following idea. Suppose that we want to prove that a statement s is true. Instead of showing directly that the truth value of s is T , we assume that s is false; in other words, we assume that :s is true. Our aim is to produce a valid mathematical argument based on the assumed truth of the premise :s which leads to some statement r that is false. The resulting conditional statement (:s) ) r is based on a valid mathematical deduction so it is a true statement, as explained in Remark 2.2.12. As such, (:s) ) r can only have one of the forms T ) T ',F ) T ' or F ) F '. Given that the truth value of r is F , we conclude that our statement (:s) ) r has the formF ) F '. In this way we deduce that the truth value of the premise :s is F and hence the truth value of s is T!
In the special case where the statement s whose truth we want to establish has the conditional form p ) q', we need to assume thatp ) q' is false; that is, we need to assume that :(p ) q)' is true. Using the fact established in subsection 2. thatp ) q' is logically equivalent to (:p) _ q', we see that:(p ) q)' is logically equivalent to `:
(:p)_q
'. In turn, this is logically equivalent to p^(:q)' by one of the de Morgan's laws. Thus, in order to prove by contradiction thatp ) q' is true, we need to assume that `p ^ (:q)' is true (equivalently, that p is true and q is false) and then produce a valid mathematical argument that leads to some statement r that is false.
Let us apply this idea to the universal statement encountered in Example 2.5.2 in order to prove its validity by contradiction.
Example 2.5.5 Show that for all integers n, if n^2 is even, then n is even.
This is a conditional statement of the form p ) q' where p is the predicaten^2 is even' and q is the predicate n is even'. As explained above, in order to establish the truth of this conditional statement by contradiction, we need to assume thatn^2 is even' and `n is not even' and produce a valid mathematical argument that leads to some statement r that is false.
We rst use the fact that n is an integer, so if it not even, then it is odd. Therefore, we assume that n^2 is even' andn is odd', and our aim is to reach a contradiction in the form of some false statement r.
Assuming that n is odd, let n = 2s+1 for some integer s. Based on that assumption, we see that n^2 = (2s + 1)^2 = 4s^2 + 4s + 1 = 2(2s^2 + 2s) + 1 is of the form 2k + 1 where k = 2s^2 + 2s is an integer, so n^2 is odd. However, we have also assumed that n^2 is even. Thus, we have reached the statement r that `n^2 is both odd and even' which is clearly false. Our task has been completed.
2.6 Practice questions
Practice question 2.6.1 Use a truth table to prove that the propositions
q,
(p ^ q) _ q, and (p _ q) ^ q are all logically equivalent.
Practice question 2.6.2 For integers n, consider the universal statements
8 even n [(2n + n^3 ) is even]
and 8 odd n [(2n + n^3 ) is odd]. Show that both these statements are true.
Since we have assumed that it is difficult to derive the falsity of q directly from the given de nitions, you must follow a different approach:
(i) Start from the given de nitions and construct a valid argument which shows that the conditional proposition q ) r' is true; that is, assume the truth of q and deduce the truth of r on the basis of that assumption. Hint: Your argument may start as follows:If 1 is an even integer, then 1 = 2k for some integer k. Then, ...'.
(ii) Construct another valid argument which shows that the conditional proposition `q ) s' is also true.
It follows from parts (i) and (ii) that the conjunction (q ) r) ^ (q ) s)' is true. Hence, using the result established in Exercise 2.7.2, its logically equivalent propo- sitionq ) (r ^ s)' is true as well.
(iii) Given that the conditional q ) (r ^ s)' is true and thatr ^ s' is absurd and hence false, what do you infer about the truth value of q?
Note that this approach amounts to proving by contradiction that the proposition 1 is not an even integer' is true. Indeed, you have assumed that1 is not an even integer' is false (equivalently, you have assumed that 1 is an even integer' is true) and reached the absurd conclusion that2 is both an odd and an even integer'.
Exercise 2.7. For integers n, consider the universal statement ` 8 n [(n^2 + 3n) is odd]'.
(i) Prove that this statement is false.
For integers n, consider the existential statement ` 9 n [(n^2 + 3n) is odd]'.
(ii) Use the de nitions given in Exercise 2.7.3 to prove that this statement is false as well. Hint: Prove instead that its negation is true.
2.8 Exam-style question
(i) De ne what the terms proposition, predicate and existential statement mean.
(ii) Write down the truth table of the conditional proposition p ) q' and the truth table of its converse propositionq ) p'. Assuming that you know that
the proposition `q ) p' is false, establish the truth value of each of the following propositions:
p ) q, (:p) ) (:q), (:q) ) (:p).
(iii) For integers n, write down the negation of the existential statement
9 n [n^3 = 16]
and hence prove that the statement ` 9 n [n^3 = 16]' is false.
For the next two parts, you will need the following de nitions:
An integer n is even only if n = 2k for some integer k.
An integer n is odd only if n = 2k + 1 for some integer k.
(iv) For integers n, prove by contraposition that the conditional statement `if n is not even, then n + 1 is not odd' is true.
(v) For integers n, prove by contradiction that the conditional statement `if n is not odd, then n + 1 is not even' is true.
2.9 Relevant sections from the textbooks
M. Anthony and M. Harvey, Linear Algebra, Concepts and Methods, Cambridge University Press.
Some paragraphs on logic can be found on pages 8 and 9 of our Algebra Textbook as part of its preliminaries. I would also recommend that you read through the entire preliminaries, from page 1 to page 9.
N. Biggs, Discrete Mathematics, Oxford University Press.
If you have a copy of this book, chapters 1 and 3 can be useful as additional reading.
