1. **State the problem:** We need to prove the equivalence of the expressions using truth tables. Expression a) $AC + BC = (A + B)C$ Expression b) $ABC = A + B + C$ 2. **Recall the logic operations:** - $+$ means OR - Concatenation (e.g., $AC$) means AND - Equality means both sides have the same truth value for all inputs 3. **Construct truth tables for expression a):** | A | B | C | AC | BC | AC + BC | A + B | (A + B)C | |---|---|---|----|----|---------|-------|----------| | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | Both columns $AC + BC$ and $(A + B)C$ match exactly, proving equivalence. 4. **Construct truth tables for expression b):** | A | B | C | ABC | A + B + C | |---|---|---|-----|-----------| | 0 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 1 | | 0 | 1 | 0 | 0 | 1 | | 0 | 1 | 1 | 0 | 1 | | 1 | 0 | 0 | 0 | 1 | | 1 | 0 | 1 | 0 | 1 | | 1 | 1 | 0 | 0 | 1 | | 1 | 1 | 1 | 1 | 1 | The columns $ABC$ and $A + B + C$ do not match for all inputs, so these expressions are not equivalent. **Final conclusion:** - Expression a) is true: $AC + BC = (A + B)C$ - Expression b) is false: $ABC \neq A + B + C$