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Exercises

Use universal instantiation or universal modus ponents to complete the following arguments.

  1. If $a, b, c, d \in \mathbb{Z}$ and $b, d \ne 0$ then $\frac{a/b}{c/d} = \frac{ad}{bc}$
    $a=2, b=3, c=6, d=2$
    ∴ ___________________________green A - final answer

  1. All dogs can swim.
    John has a dog named Solar.
    ∴ ___________________________

  1. Differentiable functions are continuous.
    $f(x) = x^2 + 2x - 3$ is differentiable
    ∴ ___________________________green A - final answer

  1. If the lengths of the sides of a triangle satisfy the equation $a^2 + b^2 = c^2$ then the triangle is a right triangle.
    The sides of a triangle are 5, 12 and 13.
    ∴ ___________________________

The following arguments are either valid or invalid. The valid arguments are examples of universal modue ponents or universal modus tollens; the invalid arguments are either converse errors or inverse errors. Determine which are valid or invalid and why.

  1. Every integer greater than 1 is divisible by a prime number.
    14 is an integer greater than 1.
    ∴ 14 is divisible by a prime number.green video - video solution

  1. All rectangles have four sides.
    R is not a rectangle.
    R doesn't have four sides.

  1. If $f:X\to Y$ and $g:Y\to Z$ are both one-to-one functions then $g \circ f$ is one-to-one.
    g is not one-to-one.
    ∴ $g \circ f$ is not one-to-one.green question mark - hintgreen check mark - show solution

  1. All birds can fly.
    John's pet can fly.
    ∴ John's pet is a bird.

  1. All even numbers are divisible by 2.
    7 is not divisible by 2.
    ∴ 7 is not even.green check mark - show solution

  1. If a function is differentiable then it's continuous.
    $f(x) = |x|$ is continuous.
    ∴ $f(x)$ is differentiable.

  1. If G is a connected graph with integer n vertices and $n - 1$ edges then G is a tree.
    G is a graph with n vertices and is not a tree.
    G doesn't have $n - 1$ vertices or is not connected.green video - video solution

  1. If x and y are real numbers and $xy=0$ then $x=0$ or $y=0$.
    $x=2$ and $xy=0$.
    ∴ $y=0$.

Determine whether or not each of the following arguments is valid using a diagram.

  1. All polynomials are continuous.
    $f(x)$ isn't a polynomial.
    ∴ $f(x)$ isn't continuous.green video - video solution

  1. All tall people are wealthy
    All wealthy people are admired.
    Some basketball players are not admired.
    ∴Some basketball players are not tall.

  1. All dogs can swim.
    No wolves can swim.
    ∴ No wolves are dogs.green video - video solution

  1. All poets are young.
    Some young people are wise.
    John is a young poet.
    ∴ John is not a wise.

  1. No restaurants serve fish.
    Fish is never expensive.
    ∴ Some restaurants serve expensive food.green video - video solution

  1. All widgets are smooth.
    Some widgets are round.
    A round objects are green.
    ∴ No round objects are smooth.

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