Identities

An identity is an equality that is always true. It holds for every valid value of its variables - or, if there are no variables involved, it is logically true by its very nature.

There are two main types of identities: numerical and mathematical.

Numerical identities are equalities that are always true between fixed numbers (e.g., $2 = 2$).

Mathematical identities, by contrast, involve variables and remain valid for all permissible values of those variables (e.g., $(x + 1)^2 = x^2 + 2x + 1$).

Note. Beyond these, there are many other types of identities - logical, functional, trigonometric, complex, and more. However, for now, we'll narrow our focus to the two most commonly encountered in algebra, so we can examine them more thoroughly.

Here’s a classic example of an identity:

$$ (x+y)^2 = x^2 + 2xy + y^2 $$

This equality holds for all possible values of the variables x and y. No matter what numbers you plug in, both sides of the equation will always be equal.

For example, let’s set x = 1 and y = 2. We get the same result on both sides:

$$ (1+2)^2 = 1^2 + 2 \cdot 1 \cdot 2 + 2^2 $$

$$ 3^2 = 1 + 4 + 4 $$

$$ 9 = 9 $$

So the identity is verified in this case - and it will continue to hold for any other real values of x and y.

Note. In an identity, the expression on the left side of the equals sign is called the left-hand side (LHS), while the expression on the right is referred to as the right-hand side (RHS).

Numerical Identity

A numerical identity is a statement of equality between two numbers or two variable-free expressions whose values are identical by definition or direct computation.

In simpler terms, it's an arithmetic truth that never changes - both sides of the equation always represent the same quantity.

Examples

The equality $5 = 5$ is self-evidently true and thus qualifies as a numerical identity.

The same applies to $3 + 2 = 7 - 2$, since both sides evaluate to 5.

Another example is $ \sqrt{49} = 7$, which holds because the square root of 49 is, by definition, 7.

Even equalities involving constants are considered numerical identities.

For instance, $\pi = \pi$ is valid because $ \pi $ represents a constant value and is always equal to itself.

What are they used for? Numerical identities confirm established equalities, serve as foundational truths in proofs, and help simplify expressions. They represent unchanging facts in arithmetic. For example, when we write: $$
12 = 3 \cdot 4 $$ we're stating a numerical identity, because both sides are always equal in value.

Mathematical Identity

A mathematical identity is an equality involving one or more variables that holds true for all values within the domain where both sides are defined.

In other words, unlike a general equation - which is true only for specific values (its solutions) - an identity behaves like a universal rule: it's always valid, except where either expression becomes undefined (such as division by zero).

Mathematical identities are central to algebraic manipulation and simplification.

Example

Consider the equation:

$$ 2(x + 3) = 2x + 6 $$

This is a mathematical identity because no matter which real number we substitute for $x$, both sides yield the same result.

For instance, if $x = 4$:

• $2(x+3) = 2(4 + 3) = 2 \cdot 7 = 14$
• $2x+6 = 2 \cdot 4 + 6 = 8 + 6 = 14$

The identity holds for $x = -2$, $x = 0$, $x = \pi$, and so on - it’s universally true within its domain.

Example 2

Here's an identity with a restricted domain:

$$ \frac{1}{x^2 - 2x + 1} = \frac{1}{(x - 1)^2} $$

This equality is valid for all real numbers except $x = 1$, where the denominator becomes zero and the expression is undefined.

Thus, it’s a valid identity over the domain $ \mathbb{R} \setminus \{1\} $.

Example 3

The square of a binomial is another classic identity:

$$ (x+3)^2 = x^2 + 6x + 9 $$

This identity holds for every real number.

Example 4

A more advanced example: $\sin^2 x + \cos^2 x = 1$ is a fundamental identity in trigonometry.

It holds for all real values of $x$.

Note. Another key trigonometric identity is $\tan x = \frac{\sin x}{\cos x}$, which is valid wherever the expression is defined - that is, for all $x \in \mathbb{R} \setminus \left\{ \frac{\pi}{2} + k\pi \,|\, k \in \mathbb{Z} \right\}$, where the denominator is not zero. This is known as a conditional identity, valid only within its domain.

How to Determine Whether an Equality Is an Identity

Method 1: Testing with Specific Values

One way to explore whether an equality is an identity is to substitute specific values for the variables and compare the results on both sides. If the outcomes match for every value tested, the equality might be an identity.

For instance, consider the following equation:

$$ (x+y)^2 = x^2 + 2xy + y^2 $$

Let’s substitute x = 2 and y = 3 into both sides and simplify:

$$ (2+3)^2 = 2^2 + 2 \cdot 2 \cdot 3 + 3^2 $$

$$ 5^2 = 4 + 12 + 9 $$

$$ 25 = 25 $$

Since both sides yield the same value, the equality appears valid for these values.

However, to confirm that it's truly an identity, this result would need to hold for all possible values of x and y.

Note. This approach requires checking an infinite number of combinations - something clearly unfeasible in practice. That’s why this method is helpful for building intuition, but it cannot serve as a formal proof.

Method 2: Algebraic Verification

An equality is an identity if the difference between the two sides simplifies to zero, regardless of the variable values (within the domain of definition).

Let’s revisit the same equation:

$$ (x+y)^2 = x^2 + 2xy + y^2 $$

We subtract the right-hand side from both sides:

$$ (x+y)^2 - (x^2 + 2xy + y^2) = (x^2 + 2xy + y^2) - (x^2 + 2xy + y^2) $$

$$ (x+y)^2 - x^2 - 2xy - y^2 = 0 $$

Now simplify the left-hand side:

$$ x^2 + 2xy + y^2 - x^2 - 2xy - y^2 = 0 $$

$$ \require{cancel} \cancel{x^2} + \cancel{2xy} + \cancel{y^2} - \cancel{x^2} - \cancel{2xy} - \cancel{y^2} = 0 $$

$$ 0 = 0 $$

The expression simplifies to zero, confirming that the equality is indeed an identity.

This shows the equality holds for all real values of x and y, as long as the expressions are defined.

Domain Restrictions and Conditions of Validity

When dealing with algebraic fractions, it's essential to consider the domain of definition - that is, the set of values for which the expressions are actually defined.

If one of the expressions involved becomes undefined for certain values (e.g., due to division by zero), then the identity does not hold at those points.

Consider the identity:

$$ \frac{xy}{y} = x $$

The left-hand side is undefined when \( y = 0 \), since division by zero is not allowed.

Therefore, the identity is valid only when \( y \ne 0 \).

In this case, the condition of existence (domain restriction) is:

$$ C.E.: \ y \ne 0 $$

Difference Between Identities and Equations

Example 1

The following equality is an equation, not an identity, because it is true only when \( x = 5 \) or \( x = -5 \):

$$ x^2 = 25 $$

Since it does not hold for every possible value of \( x \), it cannot be considered an identity.

Note. In algebra, every identity is also an equation, but not every equation is an identity. For example, $$ x^2 = x \cdot x $$ is both an equation and an identity, because it holds for all real values of \( x \). In contrast, $$ 2x = x \cdot x $$ is an equation, but not an identity - it is only satisfied for specific values of \( x \). It's worth noting, however, that this distinction applies specifically to algebraic identities. There are many other kinds of identities - such as logical identities - that are conceptually different and not derived from equations.

Logical Identities

A logical identity is a formula in propositional logic that is always true, no matter what truth values are assigned to its component propositions.

In essence, a logical identity is a type of tautology: a compound statement that evaluates to true under every possible assignment of truth values to its variables.

Logical identities serve in logic the same role that algebraic identities serve in algebra: they are universally valid rules that allow expressions to be rewritten or transformed without altering their logical content.

Here are several fundamental logical identities:

• Idempotent Laws
  Repeating the same proposition with “or” or “and” does not affect its truth value: $$ p \lor p \equiv p \qquad \text{and} \qquad p \land p \equiv p $$
• Double Negation
  Negating a proposition twice returns the original proposition: $$ \neg(\neg p) \equiv p $$
• De Morgan’s Laws
  The negation of a conjunction is equivalent to the disjunction of the negations, and vice versa: $$ \neg(p \land q) \equiv \neg p \lor \neg q $$ $$ \neg(p \lor q) \equiv \neg p \land \neg q $$
• Implication as Disjunction
  The statement “if p, then q” is logically equivalent to “not p or q”: $$ p \Rightarrow q \equiv \neg p \lor q $$

Purpose: Logical identities are essential for simplifying logical expressions, constructing formal proofs, and verifying the validity of arguments within deductive systems.

Additional Notes

Here are some further insights about identities:

• Not all indeterminate equations (i.e., those with infinitely many solutions) are identities
  An indeterminate equation has infinitely many solutions, but that doesn't automatically make it an identity. For example, consider: $$ |x| = x $$ This holds for all \( x \ge 0 \), but it fails for values like \( x = -2 \), where we get: $$ |{-2}| = 2 \quad \text{but} \quad -2 \ne 2 $$ Hence, the equality is not universally valid and cannot be considered an identity.
• The distinction between equality and identity
  An equality is a general mathematical relation - it may or may not be true, depending on the context. An identity, by contrast, expresses a structural truth: it is always valid wherever the expressions involved are defined.
  • Equality: States that two expressions have the same value under certain conditions. It may be true or false. For example: \( x^2 = 4 \) is true only when \( x = \pm 2 \).
  • Identity: An equality that holds for all admissible values of the variables, typically as a result of algebraic structure or definition. For example: \( x^2 = x \cdot x \).

And so on.