Page 18 - 'Blast_Into_Math

Blast into Math! Pure mathematics: the proof of the pudding is in the eating

and

x =2 =⇒ 2+ x =4,

which we can write as

2+ x =4 ⇐⇒ x =2,

because the implication is true in both directions.

Definition 2.3.3. The statements A and B are equivalent if both A =⇒ B and B =⇒ A , and we write

A ⇔ B.

In plain English two statements are equivalent if they have the same meaning. Another way to say that
the statements A and B are equivalent is:

A if and only if B,

which is often abbreviated

A iff B.

It is often not obvious that two statements have the same meaning. For example, to see that

2+ x =4 ⇐⇒ x =2,

we should check both directions:

1. 2+ x =4 implies x =2, and
2. x =2 implies 2+ x =4.

To verify that two statements are equivalent we should always carefully check that both directions are true.

When a statement is false, then its negation is true.

Definition 2.3.4. The negation of a statement is a statement which has the opposite meaning. For a statement
A, we write “not A” to indicate the negation of A. The negation of a statement is the statement that must
be true if the original statement is false.

Let’s practice negating statements.

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