Logic manipulation exercises

Q1. Convert the expression below into its contrapositive.

Q2. Negate the expression below.

Q3. The expression below states `Every real number has a cube root'.
Modify the statement to say that `Every real number has a unique cube root'.

Q4. A student tried to write the statement `if a real number is positive then its cube is positive' in mathematical notation as follows: \[ \forall x > 0 \implies x^3 > 0 \] Is this correct? If not, what is wrong?
Hint: could you construct the above using blocks?
Construct a valid expression for this statement using blocks below.

Q5. Recall that a sequence \(f\) is nondecreasing if \( \forall n \in \mathbb{N} \;\; f(n+1) \geq f(n) \); it is nonincreasing if \( \forall n \in \mathbb{N} \;\; f(n+1) \leq f(n) \); it is monotonic if it is nondecreasing or nonincreasing. A student tried to write `\(f\) is monotonic' using blocks as below. What is wrong with this? Fix it by constructing a correct expression for `\(f\) is monotonic'.

Q6. Consider the proverb "All that glitters is not gold ". A student expressed this using blocks as shown below.
Is this 'correct'? Does it

• reflect the literal meaning of the sentence?
• capture the intended meaning of the proverb? If not, modify the blocks so that it does.