Following our normal procedure to solve this type of exercise, we start with the definitions. Definition: “A function f is said to be one-to-one, or an injunction, if and only if f (a) = f (b) implies that a = b for all a and b in the domain of f. A function is said […]
Following our normal procedure to solve this type of exercise, we start with the definitions. Definition: “A function f is said to be one-to-one, or an injunction, if and only if f (a) = f (b) implies that a = b for all a and b in the domain of f. A function is said
Determine whether each of these functions from Z to Z is one-to-one Read More »
Following our normal procedure to solve this type of exercise, we start with the definitions. Definition: “A function f from A to B is called onto, or a surjection, if and only if for every element b∈B there is an element a∈A with f(a)=b. A function f is called surjective if it is onto.” Discrete
Which functions in Exercise 12 are onto? Read More »
Note that in each case, to find the domain, determine the set of elements assigned values by the function. As you should know by now, we need to always start with the definitions. Definition: “If f is a function from A to B, we say that A is the domain of f and B is
Find the domain and range of these functions Read More »
When solving exercises related to functions, a common type of problem is to find out if what we are given is actually a function or not. As with almost everything in mathematics, we need to start with the definition. Definition: “Let A and B be nonempty sets. A function f from A to B is
Why is f not a function from R to R if Read More »
In this case, we will attempt a proof by contradiction. Remember that to use proof by contradiction to prove a conditional statement(p->q), we do the following: This type of proofs is valid because p->q ≡ (p^q)-> F. p= “mn is even” q= “m is even or n is even” ¬q = “m and n are
Prove that if m and n are integers and mn is even, then m is even or n is even Read More »
This type of exercise tends to be easier, as it tells you and advance what method you should use. Remember the beauty of proving a theorem, is that you have an arsenal of methods to use and most of the time, your skill identifying what method to use is what will take you to a
One of the easiest methods to proof that conjecture is false is to use a counterexample. So, in this case, we will try to find a counterexample. We know that √2 is an irrational number. √2 x √2 = 2 The number 2 is a rational number. Therefore, the product of two irrational numbers is
Prove or disprove that the product of two irrational numbers is irrational Read More »
To determine whether an argument is valid or not, we should prove that it is impossible that all the premises are true and the conclusion is false. The way we do that proof is by applying one or more rules of inference on the premises and obtaining the conclusion. Like in other exercises, it is
Like in other exercises related to rules of inferences, you first should know the basic rules of inferences before you can apply them to draw conclusions from certain premises. If you still don’t know the rules of inference, make sure you open your textbook and have rules at hand so you can apply them. Notice
For each of these arguments, explain which rules of inference are used for each step Read More »
