1. Problem 7(a): Identify the law for $A+AB+ABC+ABCD=ABCD+ABC+AB+A$. Step 1: Notice the left side is $A + AB + ABC + ABCD$. Step 2: Using the **Commutative Law** of addition, terms can be reordered: $A + AB + ABC + ABCD = ABCD + ABC + AB + A$. Step 3: This equality is based on the **Commutative Law** of Boolean addition. 2. Problem 7(b): Identify the law for $A+AB+ABC+ABCD=DCBA+CBA+BA+A$. Step 1: The right side is $DCBA + CBA + BA + A$ which is the reverse order of the left side terms. Step 2: Since Boolean multiplication is commutative, $ABCD = DCBA$, $ABC = CBA$, and $AB = BA$. Step 3: This equality uses the **Commutative Law** of Boolean multiplication and addition. 3. Problem 7(c): Identify the law for $AB(CD+CD+EF+EF) = ABCD + ABCD + ABEF + ABEF$. Step 1: Inside the parentheses, $CD+CD+EF+EF$ simplifies to $CD + EF$ by the **Idempotent Law** ($X+X=X$). Step 2: Distribute $AB$ over $CD + EF$: $ABCD + ABEF$. Step 3: The right side has repeated terms $ABCD + ABCD + ABEF + ABEF$, which simplifies to $ABCD + ABEF$ by the **Idempotent Law**. Step 4: This equality is based on the **Distributive Law** and **Idempotent Law**. 4. Problem 8(a): Identify the rule for $AB + CD + EF = AB + CD + EF$. Step 1: Both sides are identical. Step 2: This is the **Identity Law** or simply equality. 5. Problem 8(b): Identify the rule for $AAB + ABC + ABB = ABC$. Step 1: Simplify $AAB = AB$ (since $AA = A$ by **Idempotent Law**). Step 2: Simplify $ABB = AB$. Step 3: So left side becomes $AB + ABC + AB$. Step 4: $AB + ABC = AB$ by the **Absorption Law**. Step 5: Therefore, left side simplifies to $AB$. Step 6: The right side is $ABC$. Step 7: Since $ABC$ is a subset of $AB$, the equality $AB + ABC + AB = ABC$ is not generally true unless $AB = ABC$. Step 8: Possibly a typo; assuming the intended equality is $AAB + ABC + ABB = AB$. Step 9: The simplification uses **Idempotent Law** and **Absorption Law**. 6. Problem 8(c): Identify the rule for $A(BC + BC) + AC = A(BC) + AC$. Step 1: Inside parentheses, $BC + BC = BC$ by **Idempotent Law**. Step 2: So left side is $A(BC) + AC$. Step 3: Both sides are equal. Step 4: This uses the **Idempotent Law**. 7. Problem 8(d): Identify the rule for $AB(C + C) + AC = AB + AC$. Step 1: Inside parentheses, $C + C = C$ by **Idempotent Law**. Step 2: So left side is $AB imes C + AC = ABC + AC$. Step 3: Since $ABC o AB$ when $C=1$, but generally $ABC + AC = AC$ if $AB o A$. Step 4: Actually, $AB(C + C) = ABC$. Step 5: The equality $ABC + AC = AB + AC$ is not generally true. Step 6: Possibly the intended equality is $AB(C + C) + AC = AB + AC$. Step 7: Using $C + C = C$, left side is $ABC + AC$. Step 8: Using **Absorption Law**, $ABC + AC = AC$. Step 9: So $AB + AC$ on right side is not equal to $AC$ unless $AB o AC$. Step 10: Assuming typo, the law used is **Distributive Law** and **Idempotent Law**. 8. Problem 8(e): Identify the rule for $AB + ABC = AB$. Step 1: $AB + ABC = AB$ by the **Absorption Law**. 9. Problem 8(f): Identify the rule for $ABC + AB + ABCD = ABC + AB + D$. Step 1: Left side is $ABC + AB + ABCD$. Step 2: $ABC + ABCD = ABC$ by **Absorption Law**. Step 3: So left side simplifies to $ABC + AB$. Step 4: Right side is $ABC + AB + D$. Step 5: Since $D$ is independent, equality holds only if $D$ is absorbed or redundant. Step 6: Possibly a typo; assuming intended equality is $ABC + AB + ABCD = ABC + AB$. Step 7: The law used is **Absorption Law**. Final answers: - 7(a) Commutative Law (Addition) - 7(b) Commutative Law (Multiplication and Addition) - 7(c) Distributive Law and Idempotent Law - 8(a) Identity Law - 8(b) Idempotent Law and Absorption Law - 8(c) Idempotent Law - 8(d) Distributive Law and Idempotent Law - 8(e) Absorption Law - 8(f) Absorption Law