Logic and ...

Logic¶

Task 1¶

Let \(p\), \(q\), and \(r\) be the following statements:

• \(p\): "It is raining."
• \(q\): "The sun is shining."
• \(r\): "There are clouds in the sky."

Write the following statements using logical symbols with \(p\), \(q\), \(r\), and logical operators:

1. It is raining, and the sun is shining.
2. If it is raining, then there are clouds in the sky.
3. If it is not raining, then the sun is not shining, and there are no clouds in the sky.
4. The sun is shining if and only if it is not raining.
5. If there are no clouds in the sky, then the sun is shining.

Task 2¶

Let \(p\), \(q\), and \(r\) be as defined in Task 1. Translate the following logical expressions into plain English:

1. \(p \land q \rightarrow r\)
2. \(\neg p \rightarrow q \lor r\)
3. \(\neg (p \lor q \lor r)\)
4. \((p \rightarrow r) \rightarrow q\)

Task 3¶

Provide the truth values of the statements in Tasks 1 and 2.

Task 4¶

Determine which of the following expressions are statements. Provide their truth values:

1. \(x^2 = 2\quad \forall x \in \mathbb{R}\)
2. \(x^2 = 2\) for some \(x \in \mathbb{R}\)
3. \(x^2 = x\)
4. \(x^2 = x\) for exactly one \(x \in \mathbb{R}\)
5. \(xy = z\) implies \(y = z\) for all \(x, y, z \in \mathbb{R}\)

Task 5¶

Rewrite the ambiguous expression \(x^2 = y^2\) as:

1. A precise statement whose logical value is true.
2. A precise statement whose logical value is false.

Task 6¶

Provide the contrapositive of the following statements:

1. "If I am smart, then I am rich."
2. "If \(x^2 = x\), then \(x = 0\) or \(x = 1\)."
3. "If \(2 + 2 = 4\), then \(2 + 4 = 8\)."

Task 7¶

Verify Goldbach's conjecture for small numbers, such as 6, 8, and 10. Check it for the number 98.