CNOT is a two-qubit physical state map; this first book uses only basis states. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Control off leaves zero-zero alone
CNOT is a basis-state permutation in this first book. With control off and target zero, nothing changes.
∣0,0⟩→∣0,0⟩
When the control is off, the target is unchanged
CNOT is introduced here only as a basis-state permutation. With control off, the target stays as it was.
control off⇒target unchanged
When the control is on, the target flips
With control on, the target flips. This lesson does not claim Bell states or entanglement statistics.
control on⇒target flips
Control on flips one back to zero
The fourth basis row also follows the same rule. Control one flips the target bit, so one-one maps to one-zero.
∣1,1⟩→∣1,0⟩
The four basis rows define the map here
All four rows are checked as basis-state permutations. Superposition and entanglement are deferred to later books.