1. Define two elements Boolean algebra: Boolean algebra is a mathematical structure consisting of a set with two elements, usually 0 and 1, along with two binary operations (AND \(\wedge\), OR \(\vee\)) and a unary operation (NOT \(\overline{\cdot}\)) satisfying specific axioms. 2. State and prove identity law: Identity laws state: $$a \wedge 1 = a$$ $$a \vee 0 = a$$ Proof for \(a \wedge 1 = a\): - If \(a=1\), then \(1 \wedge 1 = 1\). - If \(a=0\), then \(0 \wedge 1 = 0\). Thus, \(a \wedge 1 = a\). Similarly for \(a \vee 0 = a\). 3. Apply principle of duality: (i) Given \(a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)\) (Distributive law). Dual form: interchange \(\wedge\) and \(\vee\), and 0 and 1: $$a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$$ (ii) Given \(a \vee (a \vee b) = \overline{a} \wedge b\) (Note: original expression seems inconsistent; assuming typo, correct form is \(a \vee (a \wedge b) = a\)). Dual form: $$a \wedge (a \vee b) = a$$ 4. Students playing sports: Total students = 50 Football = 20 Hockey = 16 Both = 10 Number playing either = \(20 + 16 - 10 = 26\) Number playing neither = \(50 - 26 = 24\) 5. Define field: A field is a set with two operations (addition and multiplication) where addition forms an abelian group, multiplication forms an abelian group excluding zero, and multiplication distributes over addition. 6. State and prove De Morgan's laws: $$\overline{a \wedge b} = \overline{a} \vee \overline{b}$$ $$\overline{a \vee b} = \overline{a} \wedge \overline{b}$$ Proof for first law: - If \(a \wedge b = 1\), then \(a=1\) and \(b=1\), so \(\overline{a} = 0\) and \(\overline{b} = 0\), thus \(\overline{a} \vee \overline{b} = 0\). - If \(a \wedge b = 0\), then at least one of \(a,b\) is 0, so \(\overline{a} \vee \overline{b} = 1\). Hence, equality holds. 7. Convert expression to canonical sum of products: Given expression: \(XY + YZ\) This is already in sum of products form. 8. Relations matrices: Given $$M_R = \begin{bmatrix}0 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 0\end{bmatrix},\quad M_S = \begin{bmatrix}0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1\end{bmatrix}$$ (a) \(R \cup S\): element-wise OR (b) \(R \cap S\): element-wise AND (c) \(R \cdot S\): Boolean matrix multiplication (d) \(R \oplus S\): element-wise XOR 9. Relation \(xRy\) if \(x - y\) divisible by 5 on \(\mathbb{N}\): - Reflexive: Yes, since \(x - x = 0\) divisible by 5. - Symmetric: Yes, if \(x - y\) divisible by 5, so is \(y - x\). - Transitive: Yes, sum of multiples of 5 is multiple of 5. Thus, an equivalence relation. 10. Non-trivial subgroups of \((\mathbb{Z}_{12}, +_{12})\): Subgroups correspond to divisors of 12: - \(\{0,6\}\) - \(\{0,4,8\}\) - \(\{0,3,6,9\}\) - \(\{0,2,4,6,8,10\}\) 11. Karnaugh map minimization: Given function: $$f(a,b,c,d) = a'b'c'd' + a'b'c'd + a'b'cd' + a'bc'd' + a'bc'd + a'bcd' + abc'c'd' + abc'c'd + abcd$$ Simplify using K-map to find minimal expression. 12. Algebraic structure of \((G, *)\) with \(a * b = a + b + ab\) and \(G = \mathbb{R} - \{1\}\): 1. Closure: For \(a,b \neq 1\), \(a * b \neq 1\) (verify). 2. Associativity: Check \((a * b) * c = a * (b * c)\). 3. Identity element \(e\) satisfies \(a * e = a\). 4. Inverse of \(a\) satisfies \(a * a^{-1} = e\). 5. Commutativity: Check if \(a * b = b * a\).