Mathematical Language

Practice 1: (a) All values of $x$ less than 5.

(b) $x = 4$

(c) Both $x = 4$ and $x = 6$.

(d) $x = 4$

(e) $x = 6$ and all $x$ less than 5.

Practice 2: (a) $x + 5 < 3$.

(b) At least one prime number is even.
There is an even prime number.

(c) $x^2 ≥ 4$.

(d) $x$ does not divide 2 or $x$ does not divide 3.

(e) At least one mathematician can sing well.
There is a mathematician who can sing well.

Practice 3: Here are several ways to restate "If (a shape is a square) then (the shape is a rectangle)".
All squares are rectangles.
Every square is a rectangle.
Each square is a rectangle.
Whenever a shape is a square, then it is a rectangle.
A shape is a rectangle whenever it is a square.
A shape is a square only if it is a rectangle.
A shape is a square implies that it is a rectangle.
Being a square implies being a rectangle.

Practice 4: (a) statement "If a function is differentiable then it is continuous".
contrapositive "If a function is not continuous then it is not differentiable".

(b) statement "All men are mortal".
contrapositves "All immortals are not men".
"If a thing is not mortal then it is not human".

(c) statement "If ($x$ equals 3) then ($x^2 – 5x + 6$ equals 0)".
contrapositive "If ($x^2 – 5x + 6$ does not equal 0) then ($x$ does not equal 3)".

(d) statement "If (2 divides $x$ and 3 divides $x$) then (6 divides $x$)".
contrapositive "If (6 does not divide $x$) then (2 does not divide $x$ or 3 does not divide $x$)".