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Chapter 1
Consequence & Equivalence
This is for A Level Mathematics
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Email ID: kanwalkhalid8@gail
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Let’s look at statement 1
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There is only one prime number that is even and that is 2 so all prime numbers greater than 2 are odd
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Let’s consider statement
2
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The number 9 is an odd number but it is
not a prime number, so we cannot use this arrow <= to say that statement 2 implies statement 1

Logical equivalence 
Using arrow that points both ways
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If we
could have used both arrows (<= and =>) to indicate to say that statement 1 implies statement 2 and
statement 2 implies statement 1, then that would have been logical equivalence
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So if n2 is odd n would be odd
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So,
Statement 1Statement 2

Let’s review more examples of logical consequence and logical equivalence
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The opposite is not necessarily true
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Not just a cube
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However, statement 2 does
not imply statement 1 as x being greater than 11 could be so many values and not just 29
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However, if we start with x = -1 and then cube x, so (-1)3 would equal x
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Examples of Logical Equivalence

Considering n is a positive integer greater than 1,

Statement 1
n is a prime number

Statement 2
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n has exactly 2 factors

So n can be 2,3,5,7,11 ……
The definition of a prime number is that it has only two factors (1 and itself) so statement 1 implies
statement 2
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