Equality of Ordered Pairs

Theorem

Two ordered pairs are equal if and only if corresponding coordinates are equal:

$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$

Proof

Necessary Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs such that $\tuple {a, b} = \tuple {c, d}$.

Then $a = c$ and $b = d$.

$\Box$

Sufficient Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs.

Let $a = c$ and $b = d$.

Then:

$\tuple {a, b} = \tuple {c, d}$

$\blacksquare$

Also see

Equality of Ordered Tuples

Elements of Ordered Pair do not Commute, where it is formally noted that:

$a \ne b \iff \tuple {a, b} \ne \tuple {b, a}$

that is:

$\tuple {a, b} = \tuple {b, a} \iff a = b$

Sources

1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.7$. Pairs. Product of sets
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Introduction: Set-Theoretic Notation
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9$
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.9$: Cartesian Product
1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets: Exercise $1.4.1$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ordered pair
2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Exercise $1.1$
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ordered pair
2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom: Lemma $4.3$.