How to Write Proofs with Functions with the Math Sorcerer

Discover function basics: domain and codomain, notation, and conditions for one-to-one, onto, and bijective functions, with examples like x^2, sine, and the inverse of exponential.

Explore functions by defining domain and codomain, and distinguishing range, injective (1-to-1) and onto functions through simple examples, graphs, and proofs using the horizontal line test.

Prove that f(x) = 1/x on the nonzero real numbers is onto by showing that for every y in the codomain there exists x = 1/y with f(x) = y.

the lecture shows that f(x) = m x + b with m not zero is surjective on real numbers by solving x = (y - b)/m for y.

Prove a function from Z × Z to Z is onto by showing every y has a preimage. Choose m=0 and n=y+3, so f(0,y+3)=y.

Show that f: Z×Z → Z with f(m,n)=2m - n is onto by taking any y in Z and choosing x=(0,-y), since f(0,-y)=y.

Learn how to prove a function is onto by showing that for any y in the codomain there exists x in the domain with f(x)=y, using f(x)=1/(x-2) and x=1/y+2.

Prove that the function f(x)= sqrt(x+2) is injective by assuming f(a)=f(b), squaring to obtain a+2=b+2, and concluding a=b, demonstrating a one-to-one mapping.

Demonstrates that the function f from [0, ∞) to [0, ∞), with f(x)=x^2, is injective by showing f(a)=f(b) implies a=b for nonnegative a and b.

The lecture proves that f(n) = (-1)^n n is injective on positive integers by showing f(n) = f(m) implies n = m using the signs and exponent parity.

Learn to prove a function is not surjective by negating universal quantifiers, selecting a y in Y and showing f(x) ≠ y for all x, with f(x)=3x+2 on Z.

Prove a piecewise function is bijective by showing it is one-to-one and onto for integers to natural numbers, and derive its inverse with even and odd cases.

Demonstrate that the composition of two injective functions is injective by using the definition of injectivity and the property f∘g(a) = f∘g(b) implies a = b.

If the composition g∘f is injective, then f is injective. The proof applies g to both sides of f(x)=f(y) and uses the injectivity of g∘f to deduce x=y.

Learn how surjectivity is preserved under function composition: if g is onto from a to b and f is onto from b to c, then f∘g is onto.

Show that if g∘f is surjective, then g is surjective by picking any y in C and finding b in B with g(b)=y via f.

Prove the direct image of a union under a function: f(A∪B) = f(A) ∪ f(B) by showing both inclusions and how f maps A and B to their images.

Study the inverse image (preimage) under a function from X to Y for subsets A and B of Y, and prove that f^{-1}(A ∩ B) = f^{-1}(A) ∩ f^{-1}(B).

Learn to prove that the inverse image of an intersection equals the intersection of inverse images for a function f: X to Y, by tracking elements through domains and codomains.

Shows that for an injective function f, f(A ∩ B) = f(A) ∩ f(B) and proves it by mutual inclusion.

Learn to work with inverse images under a function, proving that the preimage of a union equals the union of preimages for sets in the codomain.

Demonstrate a careful proof that the function from two-by-two matrices to four-tuples is bijective by establishing it is injective and surjective.

Define surjective (onto) functions, showing every codomain element has a preimage, then illustrate with f: R→R, f(x)=x^2, which is not surjective because negatives are not hit.

Explore injective mappings (1 to 1) in functions and mappings, with a simple proof using F from R to R where F(x)=x+1, showing that F(x)=F(y) implies x=y.