1.1.2 Logical connectives

What you'll learn

How to use the symbols $≡\equiv$ , $⇒\Rightarrow$ and $⇔\Leftrightarrow$ correctly.
How to distinguish an identity from an equation.
How to identify the direction of an implication and write its converse.
How to prove an “if and only if” statement by establishing both directions.

Starting point: mathematical statements

A statement is a mathematical sentence that is either true or false. It is common to use letters such as $P$ and $Q$ to represent statements.

For example:

$P$ : “The integer $n$ is divisible by 4.”
$Q$ : “The integer $n$ is even.”

A logical connective is a symbol or phrase that describes the logical relationship between statements or expressions.

The three connectives in this topic have different meanings:

A comparison of identity, one-way implication and two-way equivalence

Congruence and identity: $≡\equiv$

The symbol $≡\equiv$ is read as “is identically equal to” or, in this context, “is congruent to”.

Definition

Identity

An identity is an equality that is true for every allowed value of the variable. The symbol $≡\equiv$ can be used to show that two expressions are identically equal.

For example,

(x+1)2≡x2+2x+1.(x+1)^2 \equiv x^2+2x+1.

This is true for every real value of $x$ , so it is an identity.

By contrast,

x+1)^2=9

is an equation. It is true only for particular values of $x$ , namely $x = 2$ and $x = - 4$ .

Key Idea

Identity or equation?

Use $≡\equiv$ when two expressions have the same value for every allowed value of their variables. Use $=$ when asserting equality, including when solving for particular values.

Establishing an identity

To show that two expressions are identically equal, you can transform one side using valid algebra until it matches the other side.

Example

Verifying an algebraic identity

Show that

(x+2)2−(x−2)2≡8x.(x+2)^2-(x-2)^2 \equiv 8x.

Expand both squared brackets:
$x+2)^2=x^2+4x+4$
and
$x-2)^2=x^2-4x+4.$
Substitute these expansions into the left-hand side:
$(x+2)2−(x−2)2≡(x2+4x+4)−(x2−4x+4).(x+2)^2-(x-2)^2 \equiv (x^2+4x+4)-(x^2-4x+4).$
Remove the second bracket carefully and collect like terms:
$(x2+4x+4)−(x2−4x+4)≡x2+4x+4−x2+4x−4≡8x.\begin{aligned} (x^2+4x+4)-(x^2-4x+4) &\equiv x^2+4x+4-x^2+4x-4\\ &\equiv 8x. \end{aligned}$
Therefore, the identity is established.

Common Mistake

Testing only a few values

Checking an equality for one or two values does not prove an identity. An identity must hold for every allowed value, so you need a general algebraic argument.

Implication: $⇒\Rightarrow$

The symbol $⇒\Rightarrow$ is read as “implies”.

Definition

Implication

The statement $P⇒QP\Rightarrow Q$ means “if $P$ , then $Q$ ”. Whenever $P$ is true, $Q$ must also be true.

The statement $P$ is called the hypothesis or condition, and $Q$ is called the conclusion.

For example,

even.n\text{ is divisible by }4\Rightarrow n\text{ is even}.

This is true because every integer divisible by 4 can be written as $n = 4 k = 2 (2 k)$ for some integer $k$ , so it is also divisible by 2.

Direction matters

An implication works in one stated direction. If $P⇒QP\Rightarrow Q$ is true, you cannot automatically assume that $Q⇒PQ\Rightarrow P$ is true.

The reversed statement $Q⇒PQ\Rightarrow P$ is called the converse of the original implication.

Example

Testing the converse of an implication

Consider the implication:

even.n\text{ is divisible by }4\Rightarrow n\text{ is even}.

Reverse the two statements to form the converse:
$4.n\text{ is even}\Rightarrow n\text{ is divisible by }4.$
Test the converse using an even integer such as 6. The number 6 is even because it is divisible by 2.
However, 6 is not divisible by 4. This counterexample shows that the converse is false, even though the original implication is true.

A counterexample is a single example that shows a general statement is false.

Common Mistake

Assuming the converse

From $P⇒QP\Rightarrow Q$ , you may conclude $Q$ when you know $P$ . You may not reverse the arrow unless you have separately shown that $Q⇒PQ\Rightarrow P$ .

Necessary and sufficient conditions

Implications can also be described using the words necessary and sufficient.

If $P⇒QP\Rightarrow Q$ , then:

$P$ is a sufficient condition for $Q$ : knowing $P$ is enough to guarantee $Q$ .
$Q$ is a necessary condition for $P$ : $P$ cannot be true unless $Q$ is also true.

For example, being divisible by 4 is sufficient for an integer to be even. Being even is necessary for an integer to be divisible by 4.

Tip

Translating “if” and “only if”

“ $P$ if $Q$ ” means $Q⇒PQ\Rightarrow P$ .
“ $P$ only if $Q$ ” means $P⇒QP\Rightarrow Q$ .

The condition following if points towards the result; the condition following only if is necessary.

Equivalence: $⇔\Leftrightarrow$

The symbol $⇔\Leftrightarrow$ is read as “if and only if”, often shortened to “iff”.

Definition

Logical equivalence

The statement $P⇔QP\Leftrightarrow Q$ means that both $P⇒QP\Rightarrow Q$ and $Q⇒PQ\Rightarrow P$ are true. Each statement is both necessary and sufficient for the other.

You can think of $⇔\Leftrightarrow$ as a two-way implication:

Q⇒P.P\Leftrightarrow Q \quad\text{means}\quad P\Rightarrow Q\text{ and }Q\Rightarrow P.

For example, for an integer $n$ ,

even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.

To prove this fully, you must prove both directions.

Example

Proving an if and only if statement

Prove that, for an integer $n$ ,

even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.

Prove the forward implication. If $n$ is even, then $n = 2 k$ for some integer $k$ . Therefore,
$n^2=(2k)^2=4k^2=2(2k^2),$
so $n^2$ is even.
Prove the reverse implication. Suppose $n^2$ is even. If $n$ were odd, then $n = 2 k + 1$ for some integer $k$ , giving
$n^2=(2k+1)^2=2(2k^2+2k)+1,$
which would be odd.
This contradicts the assumption that $n^2$ is even, so $n$ must be even. Both implications are true, and hence
$even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.$

Key Idea

Two directions are required

To prove $P⇔QP\Leftrightarrow Q$ , prove $P⇒QP\Rightarrow Q$ and then prove $Q⇒PQ\Rightarrow P$ . Proving only one direction establishes an implication, not an equivalence.

Using connectives in algebra

Logical symbols are useful when presenting chains of reasoning. However, the symbol you choose must accurately describe the relationship between consecutive lines.

Consider solving $x^2=16$ over the real numbers:

x=−4.\begin{aligned} x^2=16 &\Leftrightarrow x^2-16=0\\ &\Leftrightarrow (x-4)(x+4)=0\\ &\Leftrightarrow x=4\text{ or }x=-4. \end{aligned}

Each line is equivalent to the previous line, so the double implication is appropriate.

By contrast,

x=4⇒x2=16x=4\Rightarrow x^2=16

is true, but

x2=16⇒x=4x^2=16\Rightarrow x=4

is false because $x = - 4$ is another possibility. Therefore, $x=4⇔x2=16x=4\Leftrightarrow x^2=16$ would be incorrect.

Common Mistake

Losing solutions in an equivalence chain

A step is not reversible if it removes possible solutions. For example, replacing $x^2=16$ with $x = 4$ loses the solution $x = - 4$ , so those statements are not equivalent.

Exam technique

In the exam

Decide whether the relationship is an identity, a one-way consequence or a two-way equivalence before choosing $≡\equiv$ , $⇒\Rightarrow$ or $⇔\Leftrightarrow$ .
When asked to prove an “if and only if” statement, clearly establish both implications, preferably in separate parts.
Test a suspected converse with simple values; one valid counterexample is enough to disprove a universal claim.
Check that every $⇔\Leftrightarrow$ step in algebra can be reversed without introducing or losing solutions.

Self review

Check yourself

What is the difference between $(x+1)2≡x2+2x+1(x+1)^2\equiv x^2+2x+1$ and $x+1)^2=9$ ?
If $P⇒QP\Rightarrow Q$ , what is the converse, and must it be true?
What two separate implications must you prove to establish $P⇔QP\Leftrightarrow Q$ ?

What you'll learn

How to use the symbols $≡\equiv$ , $⇒\Rightarrow$ and $⇔\Leftrightarrow$ correctly.
How to distinguish an identity from an equation.
How to identify the direction of an implication and write its converse.
How to prove an “if and only if” statement by establishing both directions.

Starting point: mathematical statements

A statement is a mathematical sentence that is either true or false. It is common to use letters such as $P$ and $Q$ to represent statements.

For example:

$P$ : “The integer $n$ is divisible by 4.”
$Q$ : “The integer $n$ is even.”

A logical connective is a symbol or phrase that describes the logical relationship between statements or expressions.

The three connectives in this topic have different meanings:

A comparison of identity, one-way implication and two-way equivalence

Congruence and identity: $≡\equiv$

The symbol $≡\equiv$ is read as “is identically equal to” or, in this context, “is congruent to”.

Definition

Identity

An identity is an equality that is true for every allowed value of the variable. The symbol $≡\equiv$ can be used to show that two expressions are identically equal.

For example,

(x+1)2≡x2+2x+1.(x+1)^2 \equiv x^2+2x+1.

This is true for every real value of $x$ , so it is an identity.

By contrast,

x+1)^2=9

is an equation. It is true only for particular values of $x$ , namely $x = 2$ and $x = - 4$ .

Key Idea

Identity or equation?

Use $≡\equiv$ when two expressions have the same value for every allowed value of their variables. Use $=$ when asserting equality, including when solving for particular values.

Establishing an identity

To show that two expressions are identically equal, you can transform one side using valid algebra until it matches the other side.

Example

Verifying an algebraic identity

Show that

(x+2)2−(x−2)2≡8x.(x+2)^2-(x-2)^2 \equiv 8x.

Expand both squared brackets:
$x+2)^2=x^2+4x+4$
and
$x-2)^2=x^2-4x+4.$
Substitute these expansions into the left-hand side:
$(x+2)2−(x−2)2≡(x2+4x+4)−(x2−4x+4).(x+2)^2-(x-2)^2 \equiv (x^2+4x+4)-(x^2-4x+4).$
Remove the second bracket carefully and collect like terms:
$(x2+4x+4)−(x2−4x+4)≡x2+4x+4−x2+4x−4≡8x.\begin{aligned} (x^2+4x+4)-(x^2-4x+4) &\equiv x^2+4x+4-x^2+4x-4\\ &\equiv 8x. \end{aligned}$
Therefore, the identity is established.

Common Mistake

Testing only a few values

Checking an equality for one or two values does not prove an identity. An identity must hold for every allowed value, so you need a general algebraic argument.

Implication: $⇒\Rightarrow$

The symbol $⇒\Rightarrow$ is read as “implies”.

Definition

Implication

The statement $P⇒QP\Rightarrow Q$ means “if $P$ , then $Q$ ”. Whenever $P$ is true, $Q$ must also be true.

The statement $P$ is called the hypothesis or condition, and $Q$ is called the conclusion.

For example,

even.n\text{ is divisible by }4\Rightarrow n\text{ is even}.

This is true because every integer divisible by 4 can be written as $n = 4 k = 2 (2 k)$ for some integer $k$ , so it is also divisible by 2.

Direction matters

An implication works in one stated direction. If $P⇒QP\Rightarrow Q$ is true, you cannot automatically assume that $Q⇒PQ\Rightarrow P$ is true.

The reversed statement $Q⇒PQ\Rightarrow P$ is called the converse of the original implication.

Example

Testing the converse of an implication

Consider the implication:

even.n\text{ is divisible by }4\Rightarrow n\text{ is even}.

Reverse the two statements to form the converse:
$4.n\text{ is even}\Rightarrow n\text{ is divisible by }4.$
Test the converse using an even integer such as 6. The number 6 is even because it is divisible by 2.
However, 6 is not divisible by 4. This counterexample shows that the converse is false, even though the original implication is true.

A counterexample is a single example that shows a general statement is false.

Common Mistake

Assuming the converse

From $P⇒QP\Rightarrow Q$ , you may conclude $Q$ when you know $P$ . You may not reverse the arrow unless you have separately shown that $Q⇒PQ\Rightarrow P$ .

Necessary and sufficient conditions

Implications can also be described using the words necessary and sufficient.

If $P⇒QP\Rightarrow Q$ , then:

$P$ is a sufficient condition for $Q$ : knowing $P$ is enough to guarantee $Q$ .
$Q$ is a necessary condition for $P$ : $P$ cannot be true unless $Q$ is also true.

For example, being divisible by 4 is sufficient for an integer to be even. Being even is necessary for an integer to be divisible by 4.

Tip

Translating “if” and “only if”

“ $P$ if $Q$ ” means $Q⇒PQ\Rightarrow P$ .
“ $P$ only if $Q$ ” means $P⇒QP\Rightarrow Q$ .

The condition following if points towards the result; the condition following only if is necessary.

Equivalence: $⇔\Leftrightarrow$

The symbol $⇔\Leftrightarrow$ is read as “if and only if”, often shortened to “iff”.

Definition

Logical equivalence

The statement $P⇔QP\Leftrightarrow Q$ means that both $P⇒QP\Rightarrow Q$ and $Q⇒PQ\Rightarrow P$ are true. Each statement is both necessary and sufficient for the other.

You can think of $⇔\Leftrightarrow$ as a two-way implication:

Q⇒P.P\Leftrightarrow Q \quad\text{means}\quad P\Rightarrow Q\text{ and }Q\Rightarrow P.

For example, for an integer $n$ ,

even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.

To prove this fully, you must prove both directions.

Example

Proving an if and only if statement

Prove that, for an integer $n$ ,

even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.

Prove the forward implication. If $n$ is even, then $n = 2 k$ for some integer $k$ . Therefore,
$n^2=(2k)^2=4k^2=2(2k^2),$
so $n^2$ is even.
Prove the reverse implication. Suppose $n^2$ is even. If $n$ were odd, then $n = 2 k + 1$ for some integer $k$ , giving
$n^2=(2k+1)^2=2(2k^2+2k)+1,$
which would be odd.
This contradicts the assumption that $n^2$ is even, so $n$ must be even. Both implications are true, and hence
$even.n\text{ is even}\Leftrightarrow n^2\text{ is even}.$

Key Idea

Two directions are required

To prove $P⇔QP\Leftrightarrow Q$ , prove $P⇒QP\Rightarrow Q$ and then prove $Q⇒PQ\Rightarrow P$ . Proving only one direction establishes an implication, not an equivalence.

Using connectives in algebra

Logical symbols are useful when presenting chains of reasoning. However, the symbol you choose must accurately describe the relationship between consecutive lines.

Consider solving $x^2=16$ over the real numbers:

x=−4.\begin{aligned} x^2=16 &\Leftrightarrow x^2-16=0\\ &\Leftrightarrow (x-4)(x+4)=0\\ &\Leftrightarrow x=4\text{ or }x=-4. \end{aligned}

Each line is equivalent to the previous line, so the double implication is appropriate.

By contrast,

x=4⇒x2=16x=4\Rightarrow x^2=16

is true, but

x2=16⇒x=4x^2=16\Rightarrow x=4

is false because $x = - 4$ is another possibility. Therefore, $x=4⇔x2=16x=4\Leftrightarrow x^2=16$ would be incorrect.

Common Mistake

Losing solutions in an equivalence chain

A step is not reversible if it removes possible solutions. For example, replacing $x^2=16$ with $x = 4$ loses the solution $x = - 4$ , so those statements are not equivalent.

Exam technique

In the exam

Decide whether the relationship is an identity, a one-way consequence or a two-way equivalence before choosing $≡\equiv$ , $⇒\Rightarrow$ or $⇔\Leftrightarrow$ .
When asked to prove an “if and only if” statement, clearly establish both implications, preferably in separate parts.
Test a suspected converse with simple values; one valid counterexample is enough to disprove a universal claim.
Check that every $⇔\Leftrightarrow$ step in algebra can be reversed without introducing or losing solutions.

Self review

Check yourself

What is the difference between $(x+1)2≡x2+2x+1(x+1)^2\equiv x^2+2x+1$ and $x+1)^2=9$ ?
If $P⇒QP\Rightarrow Q$ , what is the converse, and must it be true?
What two separate implications must you prove to establish $P⇔QP\Leftrightarrow Q$ ?

What you'll learn

Starting point: mathematical statements

Congruence and identity: $≡\equiv$

Establishing an identity

Implication: $⇒\Rightarrow$

Direction matters

Necessary and sufficient conditions

Equivalence: $⇔\Leftrightarrow$

Using connectives in algebra

Review 1.1.2 Logical connectives by teaching Genie

1.1.2 Logical connectives

What you'll learn

Starting point: mathematical statements

Congruence and identity: $≡\equiv$

Establishing an identity

Implication: $⇒\Rightarrow$

Direction matters

Necessary and sufficient conditions

Equivalence: $⇔\Leftrightarrow$

Using connectives in algebra

Review 1.1.2 Logical connectives by teaching Genie

1.1.2 Logical connectives

What you'll learn

Starting point: mathematical statements

Congruence and identity: ≡\equiv≡

Establishing an identity

Implication: ⇒\Rightarrow⇒

Direction matters

Necessary and sufficient conditions

Equivalence: ⇔\Leftrightarrow⇔

Using connectives in algebra

Review 1.1.2 Logical connectives by teaching Genie

1.1.2 Logical connectives Revision Guide

1.1.2 Logical connectives

What you'll learn

Starting point: mathematical statements

Congruence and identity: ≡\equiv≡

Establishing an identity

Implication: ⇒\Rightarrow⇒

Direction matters

Necessary and sufficient conditions

Equivalence: ⇔\Leftrightarrow⇔

Using connectives in algebra

Review 1.1.2 Logical connectives by teaching Genie

1.1.2 Logical connectives Revision Guide

Congruence and identity: $≡\equiv$

Implication: $⇒\Rightarrow$

Equivalence: $⇔\Leftrightarrow$

Congruence and identity: $≡\equiv$

Implication: $⇒\Rightarrow$

Equivalence: $⇔\Leftrightarrow$