Visual blocks for mathematical syntax

# Logic manipulation exercises

Below are some exercises for you to try.

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.

P Q NOTR NOTQ R NOTP x y > x y > x y x y > x y x y x y = y x x y = y x z = z x = z y x > x > x n n n n n n n n n n n n x x x x x x