In topic 10 the following definitions of ’implies’ and ’implied by’ were given.
’Implies means that the first statement can be used to logically deduce the next statement. It is denoted by the symbol ’.
’Implied by means that the first statement is a logical consequence of the second statement. It is denoted by the symbol’.
The expression P Q reads ’P implies Q’.
P is called the antecedent, the premise or the hypothesis. Q is the conclusion or consequent.
There are however many other precise ways of saying exactly the same thing. The following statements are all ways of saying that P Q:
• P implies Q • if P then Q • Q follows from P • Q if P • Q is necessary for P • P is sufficient for Q
Also PQ, meaning P is implied by Q, has a similar list:
• Q implies P • if Q then P • P follows from Q • P if Q • P is necessary for Q • Q is sufficient for P
These two lists demonstrate clearly that the symbolgives the reverse implication of
the symbol
The remainder of this section will concentrate on the sign but clearly the same explanations can be found for the implication(which is much less commonly used).
The exercise which follows is similar, but not identical, to that given in topic 10. It is still worthwhile taking a few extra minutes to complete it at this stage.
Symbol exercise 5 min Æ Learning Objective
Use the correct symbol
There is an alternative version of this exercise on the web for you to try if you wish.
Q5: Insert the correct symbol between proposition A and proposition B in the following: 1. A: The liquid is colourless and clear.
B: The liquid is drinking water. 2. A: x2= -2
B: x is a complex number. 3. A: R is a square matrix.
B: The inverse matrix R-1exists. 4. A: n is an odd prime.
B: n is an odd number.
Q6: Reword the following implications using either P Q or PQ
1. If x + 8 = 4 then x is a negative number. 2. x0 is sufficient for x + 1 to be positive.
3. sin= 0 is necessary for=
The two phrases ’Q is necessary for P’ and ’P is sufficient for Q’, both of which mean P Q, give a different emphasis to the propositions.
Look at a truth table for P Q. The table is constructed by labelling the columns with the propositions P and Q and the implication P Q. The rows are then completed by considering the combinations of true (T = 1) and false (F = 0) which can occur. (T = 1 and F = 0 are known as Boolean constants.)
P Q P Q
1 1 1
1 0 0
0 0 1
0 1 1
The last two lines may seem puzzling, but can be explained by considering the meaning of P Q as ’if P then Q’. Hence if P is true, Q follows by some logical deduction: but if P is not true, Q might still be true for some other reason.
Examples
1. If P is the statement ’it is Saturday’ and Q is the statement ’it is the weekend’ then P
Q. But statement Q is true on Sundays too, when P is false.
2. If P is the statement ’x 3’ and Q is the statement ’x
2
9’ then P Q. But if, for
5.2. IMPLICATIONS 167
So when P Q either P is true - and then Q must be true - or P is false and no conclusion can be made about Q.
Hence P Q is logically equivalent to the statement ’either P is false, or Q is true, or both’ and when P is false, the implication P Q is true.
This may seem at odds with the usual sense of the word ’implication’ but it is correct within the framework of mathematical reasoning.
3. If P is the statement ’x is an odd number and 1x9’ and Q is the statement ’x is
a prime number’ then P Q.
If P is false, then Q can be either true - for example when x = 13 - or false as when x = 4 or x = 14.
However as a single mathematical proposition P Q is true.
To say ’P is sufficient for Q’ indicates that if P is false the implication is true regardless of P but that if P is true then Q has to be true for the implication to hold.
P is termed the sufficient condition.
Sufficient condition
If P Q then P is a sufficient condition for Q.
The phrase ’Q is necessary for P’ however, indicates that Q has to be true if P is true for the implication to hold but Q can be true independently of P and give a true implication. Q is termed the necessary condition.
Necessary condition
If P Q then Q is a necessary condition for P.
The same concepts apply to the lists of related phrases for P Q which were shown
earlier.
Bi-implication or equivalence The implication PQ can be read as:
• P if and only if Q • P is equivalent to Q
• P implies Q and Q implies P
The definition as given in topic 10 follows.
Equivalent to
Is equivalent to means that the first statement implies and is implied by the second
statement. (The first statement is true if and only if the second statement is true.) The truth table foris
P Q P Q
1 1 1
1 0 0
0 0 1
0 1 0
Example Let P be the statement:
A triangle has side lengths a, b and c (with c the largest) and contains a right angle. Let Q be the statement:
A triangle has side lengths a, b, c (with c the largest) and a2+ b2= c2 Then PQ
Equivalence exercise
5 min
There is a short exercise similar to this on the web for you to try if you prefer it.
Q7: State, with explanation, which of the following statements is correct with the equivalence symbol ofreplacing the question mark.
1. In the Smith family, Joe is Amy’s father ? Amy is Joe’s daughter. 2. x is divisible by 6 ? x is an even number.
3. I study Maths ? I am good at Maths. 4. x is an odd number ? x is a prime2
Negation
The symbol is used to denote negation in this topic. There are alternative symbols
such as - and ~ . The definition of negation is as follows.
Negation
• If the proposition P is true then its negationP is false.
• If the proposition P is false then its negationP is true.
Both propositions P andP cannot be true at the same time. Q8: Complete the truth table for negation.
P P
1 0
Example State the negation of the proposition P : x2
5.2. IMPLICATIONS 169
The proposition is P and the negation isP : x2
This can be checked as follows:
Let x = 4 then P is true andP is false.
Let x = 1 then P is false andP is true.
Negation exercise
5 min
There is another small exercise on the web for you to try if you wish.
Q9: For each of the propositions P, state its negationP.
1. P:
x is rational. 2. P: I like Maths.
3. P: x is a negative real. 4. P: x0, x. Inverse and converse
Returning to the original implication P Q, the inverse implication applies to the negation of both propositions P and Q.
Inverse
The inverse of the statement P Q isP Q
The converse is also worth mentioning at this point.
Converse
The converse of the statement P Q is Q P Look at the truth tables of both.
Inverse P P Q Q P Q 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 Converse Q P Q P 1 1 1 0 1 1 0 0 1 1 0 0
Notice that the results of the two truth tables are the same. That is,P Q is logically the same as Q P:
the inverse is the logical equivalent of the converse.
Example State the converse of the implication P Q where P is the proposition 1/x
and
Q is the proposition x 0
Answer:
The implication as it stands reads ’if1/x then x 0’.
The converse is Q P and reads ’ If x 0 then
1/ x ’.
Note that the converse implication does not need to hold.
Converse exercise
5 min
There is a similar web exercise for you to try if you wish.
Q10: Consider the implication S: ’if it is Thursday then I will be at home’ and state the
converse.
Q11: State the converse of the following implications:
1. I am over fifty I am over forty. 2. If it rains then it is cloudy.
3. I work only if I am at Heriot Watt University. 4. Joyce cycles to school if it is dry.
Contrapositive
The final implication to consider is the contrapositive. It is the inverse of the converse.
Since the converse and the inverse are logically equivalent, from this alone it can be deduced that the contrapositive and the original implication are logically equivalent. (It is similar to taking the inverse of an inverse in matrix algebra or reflection twice in an axis in functions).
Contrapositive
The contrapositive of the implication P Q isQ P Activity
Complete, compare and confirm that the truth tables for the implications P Q and
Q P are the same.
’When a philosopher says something which is true then it is trivial. When he says something that is not trivial then it is false.’