[Verse 1]
Start with logic, build your proof with care
Direct approach shows truth is always there
Contradiction flips the script around
Assume it's false, then watch it break down
Contrapositive switches both the sides
Mathematical induction never hides
From base case up, each step we climb
Proving statements hold for all of time
[Chorus]
Foundations first, then build it right
Logic proof and set theory's light
Functions mapping one to one
Number systems, we're never done
Count them up and break them down
Mathematical thinking sound
Foundations first, foundations true
Building math from me to you
[Verse 2]
Sets contain our elements with grace
Naive or axiomatic, find your place
ZFC axioms make it strong
Choice and comprehension sing along
Union, intersection, complement too
Power sets and empty sets breakthrough
Infinite collections in our grasp
Foundation's firm within our clasp
[Chorus]
Foundations first, then build it right
Logic proof and set theory's light
Functions mapping one to one
Number systems, we're never done
Count them up and break them down
Mathematical thinking sound
Foundations first, foundations true
Building math from me to you
[Verse 3]
Relations show how elements connect
Equivalence divides what we expect
Functions map from domain to the range
Injective, surjective, rearrange
Bijections match up perfectly
One-to-one correspondence, can't you see
From input straight to output's door
Mathematics opens up much more
[Bridge]
Natural numbers start the climb
Integers expand the line
Rationals fill gaps between
Real numbers complete the scene
Complex adds the square root dance
Of negative one's advance
Construction builds from ground up high
Each system reaching for the sky
[Verse 4]
Count the ways with permutations neat
Combinations make the picture complete
Pigeonhole says something must repeat
Inclusion-exclusion can't be beat
Prime numbers hold the secret key
Divisibility sets numbers free
Modular arithmetic goes around
Euler's theorem, wisdom found
[Chorus]
Foundations first, then build it right
Logic proof and set theory's light
Functions mapping one to one
Number systems, we're never done
Count them up and break them down
Mathematical thinking sound
Foundations first, foundations true
Building math from me to you
[Outro]
From logic gates to complex planes
Mathematical thinking breaks the chains
Foundation strong, the future's bright
Mathematical thinking takes its flight