a) Define conjunction of the statements. Prepare a truth table for the compound statement (p q) (p q) . Draw the conclusion about the statement from the truth table. [4] OR Solve the inequality 2x 1…

a) Conjunction and Truth Table for $(p\wedge q)\wedge \sim (p\vee q)$

Definition of Conjunction

A conjunction is a compound logical statement formed by combining two or more statements using the logical AND operator ($\wedge$). The conjunction $p\wedge q$ is true only when both $p$ and $q$ are true. Otherwise, it is false.

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Truth Table for $(p\wedge q)\wedge \sim (p\vee q)$

We analyze the compound statement $(p\wedge q)\wedge \sim (p\vee q)$ by constructing its truth table.

1. List all possible truth values of $p$ and $q$ (4 combinations).
2. Compute $p\wedge q$ (AND operation).
3. Compute $p\vee q$ (OR operation).
4. Compute $\sim (p\vee q)$ (NOT operation).
5. Compute $(p\wedge q)\wedge \sim (p\vee q)$ (final AND operation).

Conclusion from the Truth Table

• The final column $(p\wedge q)\wedge \sim (p\vee q)$ is always false for all possible truth values of $p$ and $q$.
• This means the statement is a contradiction (always false).
• In logical terms, $(p\wedge q)\wedge \sim (p\vee q)$ is never true under any condition.

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OR

Solution to the Inequality $|2x-1|\ge 3$ and Its Graph

Step 1: Solve the Absolute Value Inequality

The inequality $|2x-1|\ge 3$ can be split into two cases:

1. Case 1: $2x-1\ge 3$ $$\begin{gathered} 2x-1\ge 3 \\ 2x\ge 4 \\ x\ge 2 \end{gathered}$$

2. Case 2: $2x-1\le -3$ $$\begin{gathered} 2x-1\le -3 \\ 2x\le -2 \\ x\le -1 \end{gathered}$$

Final Solution

The solution set is: $$x\in (-\infty ,-1]\cup [2,\infty )$$

Graph of the Solution

The graph of $|2x-1|\ge 3$ consists of two rays:

• One extending left from $x=-1$ (including $-1$).
• One extending right from $x=2$ (including $2$).

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b) Graph of the Function $f(x)=x-x^2$ and Its Characteristics

Step 1: Rewrite the Function in Standard Quadratic Form

$$f(x)=-x^2+x$$ This is a quadratic function in the form $f(x)=a{x}^2+bx+c$, where:

• $a=-1$ (opens downward),
• $b=1$,
• $c=0$.

Step 2: Find the Vertex of the Parabola

The vertex $(h,k)$ is given by: $$h=-\frac{b}{2a}=-\frac{1}{2(-1)}=\frac{1}{2}$$ $$k=f\left(\frac{1}{2}\right)=\frac{1}{2}-{\left(\frac{1}{2}\right)}^2=\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ So, the vertex is at $\left(\frac{1}{2},\frac{1}{4}\right)$.

Step 3: Find the Roots (x-intercepts)

Set $f(x)=0$: $$\begin{gathered} x-x^2=0 \\ x(1-x)=0 \\ x=0\text{or}x=1 \end{gathered}$$ The roots are at $x=0$ and $x=1$.

Step 4: Find the y-intercept

Set $x=0$: $$f(0)=0-0^2=0$$ The y-intercept is at $(0,0)$.

Step 5: Determine the Axis of Symmetry

The axis of symmetry is the vertical line passing through the vertex: $$x=\frac{1}{2}$$

Step 6: Sketch the Graph

• The parabola opens downward (since $a=-1<0$).
• Vertex at $\left(\frac{1}{2},\frac{1}{4}\right)$.
• Roots at $x=0$ and $x=1$.
• y-intercept at $(0,0)$.

Characteristics of the Function