Use universal instantiation or universal modus ponents to complete the following arguments.
If $a, b, c, d \in \mathbb{Z}$ and $b, d \ne 0$ then $\frac{a/b}{c/d} = \frac{ad}{bc}$ $a=2, b=3, c=6, d=2$ ∴ ___________________________
All dogs can swim. John has a dog named Solar. ∴ ___________________________
Differentiable functions are continuous. $f(x) = x^2 + 2x - 3$ is differentiable ∴ ___________________________
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.
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.
All rectangles have four sides. R is not a rectangle. ∴ R doesn't have four sides.
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.
All birds can fly. John's pet can fly. ∴ John's pet is a bird.
All even numbers are divisible by 2. 7 is not divisible by 2. ∴ 7 is not even.
If a function is differentiable then it's continuous. $f(x) = |x|$ is continuous. ∴ $f(x)$ is differentiable.
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.
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.
All polynomials are continuous. $f(x)$ isn't a polynomial. ∴ $f(x)$ isn't continuous.
All tall people are wealthy All wealthy people are admired. Some basketball players are not admired. ∴Some basketball players are not tall.
All dogs can swim. No wolves can swim. ∴ No wolves are dogs.
All poets are young. Some young people are wise. John is a young poet. ∴ John is not a wise.
No restaurants serve fish. Fish is never expensive. ∴ Some restaurants serve expensive food.
All widgets are smooth. Some widgets are round. A round objects are green. ∴ No round objects are smooth.
Use universal instantiation or universal modus ponents to complete the argument
If $a, b, c, d \in \mathbb{Z}$ and $b, d \ne 0$ then $\frac{a/b}{c/d} = \frac{ad}{bc}$ $a=2, b=3, c=6, d=2$ ∴ ___________________________
$$\frac{2/3}{6/2} = \frac{2 \cdot 2}{3 \cdot 6}$$
You should stop here. This is all you can conclude from the information in this argument. Simplifying the expression would require additional information.
Use universal instantiation or universal modus ponents to complete the argument
Differentiable functions are continuous. $f(x) = x^2 + 2x - 3$ is differentiable ∴ ___________________________
$$f(x) = x^2 + 2x - 3\text{ is continuous.}$$
Notice that you don't need to know what "continuous" or "differentiable" mean since you aren't being asked to evaluate the correctness of the result. To complete the argument, all you have to do is follow the form.