The character "`=`" is commonly used to signify six different things; namely: * an equivalence relation * the standard (numeric) equality procedure * equations * identities * definitions * assignments Equivalence relations are reflexive, symmetric, and transitive binary relations. The standard (numeric) equality procedure is a procedure whose application to arguments evaluates to `#t` when all arguments are numerically equal and to `#f` otherwise. ~~~~ { .scheme } (= 1 1) ;; -> #t (= 1 2) ;; -> #f ~~~~ Equations are combinations whose head is the equals symbol, like: ~~~~ { .scheme } (= y (* 2 x)) ~~~~ Like any syntax, equations can be assigned several interpretations. One interpretation that is popular in math is to treat equations as defining relations; that is, as describing one or more sets of solutions of the equation(s). A solution is an environment in which a set of equations all evaluate to `#t`. For example, the evaluation of the following "let" expression proves that `(4, 2)` is a solution of the equation `y = 2*x`: ~~~~ { .scheme } (let ((y 4) (x 2)) (= y (* 2 x))) ;; -> #t ~~~~ Identities[^identities] are equations that have been (perhaps implicitly) universally quantified over a set of satisfying assignments. Identities are mostly used to relate values like `2`, `+`, and `*` as in: ~~~~ { .scheme } (∀ ((n ℕ)) (= (+ n n) (* 2 n))) ~~~~ [^identities]: While there are few identities written in Scheme, there are several computer algebra systems written in Scheme-like languages in which plenty of identities have been written. Definitions are compound terms whose evaluation has the side-effect of installing a new variable-to-value binding into the current environment: ~~~~ { .Scheme } (define f (λ (x) (* 2 x))) f ;; -> $