The Full Adder: Foundation of the CPU. #MTH19 #codeofalignment #education

The Full Adder: 1.Add A + B → gives: Partial Sum = A⊕B Carry₁ = AB 2.Then add: Partial Sum + Cin 3.Combine carries with OR: Cout=C1+C2 This is exactly how a full adder is physically built (two half adders + OR gate) Now let’s focus on SUM Step 1: S1=A⊕B Step 2: S=S1⊕Cin Substitute: S=(A⊕B)⊕Cin Final Result S=A⊕B⊕Cin XOR expression for Sum What I described in this video is not ordinary addition. It is: Binary addition without carry = XOR Why XOR appears Let’s check: A B A ⊕ B 0 0 0 0 1 1 1 0 1 1 1 0 Notice: •XOR = addition mod 2 •It ignores carry So what’s really happening? First stage: •XOR gives sum without carry •AND gives carry Second stage: •XOR again combines with Cin “We first combine A and B to get a partial result (XOR gives the visible sum, AND keeps the overflow). Then we combine that result with Cin, and finally merge the overflow paths.” The Sum of a Full Adder is the XOR of all inputs because XOR represents addition without carry at each stage. •XOR = “odd number of 1s” •If 1 or 3 inputs are 1 → Sum = 1 •If 0 or 2 inputs are 1 → Sum = 0 S=A⊕B⊕Cin BUT the deeper truth is: We are not just “adding twice”, we are performing carry-less addition (XOR) in stages while separately handling carry.