1. Problem A) i): Find $A \cap B$ where $A = \{1,2,3,4,5\}$ and $B = \{1,3,5,7,9\}$.\n- The intersection $A \cap B$ contains elements common to both sets.\n- Common elements are $1, 3, 5$.\n- So, $A \cap B = \{1,3,5\}$.\n\n2. Problem A) ii): Convert binary number $(11001)_2$ to decimal.\n- Use place values: $1\times 2^4 + 1\times 2^3 + 0\times 2^2 + 0\times 2^1 + 1\times 2^0$.\n- Calculate: $16 + 8 + 0 + 0 + 1 = 25$.\n- So, $(11001)_2 = 25_{10}$.\n\n3. Problem A) iii): Prove sum of first $n$ odd numbers equals $n^2$.\n- Sum of first $n$ odd numbers: $1 + 3 + 5 + \cdots + (2n-1)$.\n- Formula: $$\sum_{k=1}^n (2k-1) = n^2$$.\n- Proof by induction:\n Base case $n=1$: sum = 1, $1^2=1$, true.\n Assume true for $n$, sum = $n^2$.\n For $n+1$: sum = $n^2 + (2(n+1)-1) = n^2 + 2n + 1 = (n+1)^2$.\n- Hence proved.\n\n4. Problem A) iv): Given universal set $U=\{1,2,3,\ldots,10\}$, $A=$ primes less than 10 $=\{2,3,5,7\}$, $B=\{2,4,6,8,10\}$. Find $A \cup B$.\n- Union $A \cup B$ contains all elements in $A$ or $B$.\n- Combine: $\{2,3,4,5,6,7,8,10\}$.\n- So, $A \cup B = \{2,3,4,5,6,7,8,10\}$.\n\n5. Problem A) v): Write converse, inverse, contrapositive of "If two triangles are congruent then their sides are equal."\n- Original: If $P$ then $Q$.\n- Converse: If $Q$ then $P$. "If sides are equal then triangles are congruent."\n- Inverse: If not $P$ then not $Q$. "If triangles are not congruent then sides are not equal."\n- Contrapositive: If not $Q$ then not $P$. "If sides are not equal then triangles are not congruent."\n\n6. Problem A) vi): Determine if statement "If two triangles are congruent then their sides are equal" is tautology, contradiction, or contingency.\n- This is a true statement in geometry, always true.\n- Hence, it is a tautology.\n\n7. Problem B: Importance of mathematics in computer science.\n- Mathematics provides foundational concepts like logic, algorithms, and data structures.\n- It helps in problem-solving, designing efficient algorithms, and understanding computational complexity.\n- Essential for cryptography, graphics, machine learning, and more.\n\n"q_count":7