Question

Draw $ A^{\circ} $ ats Ask Copilot 2 of 2 ID (3 marks) b) By use of principle of mathematical induction prove that \[ 1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=n(n+1)(n+2) / 3 \] c) Use a truth table to determine whether $ \sim(p \vee(q \rightarrow r)) \leftrightarrow((p \mapsto q) \wedge r) $ a tautology. contradiction or contingency is. (3 marks) d) Given that $ f(x)=2 x+5, g(x)=x^{2}-3 $ and $ h(x)=\left\{\begin{array}{ll}x+3 & x \leq 1 \\ x-3 & x>1\end{array}\right. $. Evaluate $ h_{0} g_{a} f_{o} f_{o} g_{o} h_{o} f_{o} h(0) $ (4 marks) (2 marks) c) Show by direct proof that $ x^{2} $ is divisible by 3 , then $ x $ is divisible by 3 (2 marks) f) Negate the quantifier $ \exists x(P(x) \wedge D(x)) $ QUESTION THREE ( 20 MARKS ) a) Write the inverse, converse and contrapositive of the following statement "if $ 2+3>7 $, then cows can dance" (3 marks) b) Prove that $ \sqrt{p} $, where $ p $ is prime is irrational. (5 marks) c) Out of 130 Discrete students, 40 take coffee, 40 take cocoa, 43 take tea, 17 take cocos

          Draw
\( A^{\circ} \)
ats
Ask Copilot
2
of 2
ID
(3 marks)
b) By use of principle of mathematical induction prove that
\[
1 \cdot 2+2 \cdot 3+3 \cdot 4+\cdots+n(n+1)=n(n+1)(n+2) / 3
\]
c) Use a truth table to determine whether \( \sim(p \vee(q \rightarrow r)) \leftrightarrow((p \mapsto q) \wedge r) \) a tautology. contradiction or contingency is.
(3 marks)
d) Given that \( f(x)=2 x+5, g(x)=x^{2}-3 \) and \( h(x)=\left\{\begin{array}{ll}x+3 & x \leq 1 \\ x-3 & x>1\end{array}\right. \). Evaluate \( h_{0} g_{a} f_{o} f_{o} g_{o} h_{o} f_{o} h(0) \)
(4 marks)
(2 marks)
c) Show by direct proof that \( x^{2} \) is divisible by 3 , then \( x \) is divisible by 3
(2 marks)
f) Negate the quantifier \( \exists x(P(x) \wedge D(x)) \)

QUESTION THREE ( 20 MARKS )
a) Write the inverse, converse and contrapositive of the following statement "if \( 2+3>7 \), then cows can dance"
(3 marks)
b) Prove that \( \sqrt{p} \), where \( p \) is prime is irrational.
(5 marks)
c) Out of 130 Discrete students, 40 take coffee, 40 take cocoa, 43 take tea, 17 take cocos

$Draw A^∘ ats Ask Copilot 2 of 2 ID (3 marks) b) By use of principle of mathematical induction prove that 1 · 2+2 · 3+3 · 4+⋯+n(n+1)=n(n+1)(n+2) / 3 c) Use a truth table to determine whether ∼(p ∨(q → r)) ↔((p ↦ q) ∧ r) a tautology. contradiction or contingency is. (3 marks) d) Given that f(x)=2 x+5, g(x)=x^2-3 and h(x)={ x+3 x ≤ 1 x-3 x>1 .. Evaluate h0 ga fo fo go ho fo h(0) (4 marks) (2 marks) c) Show by direct proof that x^2 is divisible by 3 , then x is divisible by 3 (2 marks) f) Negate the quantifier ∃ x(P(x) ∧ D(x)) QUESTION THREE ( 20 MARKS ) a) Write the inverse, converse and contrapositive of the following statement "if 2+3>7, then cows can dance" (3 marks) b) Prove that √(p), where p is prime is irrational. (5 marks) c) Out of 130 Discrete students, 40 take coffee, 40 take cocoa, 43 take tea, 17 take cocos$

Added by Margaret P.

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