Discrete Mathematics: Propositions & Logic

A propositional formula that represents the statement: "Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class."

A propositional formula where the consequent (the outcome) is true only if the antecedent (the condition) is also true.

The proposition where getting an A on the final exam (p) is a necessary condition for getting an A in the class (q); if you get an A in the class, you must have gotten an A on the final.

A propositional formula that represents the statement: "To get an A in this class, it is necessary for you to get an A on the final."

An implication is false only when the antecedent is true and the consequent is false.

The antecedent is true, but the consequent is false. This makes the overall statement false.

The proposition p implies q, written as p→q, means that if p is true, then q must also be true. The truth table for implication shows that the only case where p→q is false is when p is true and q is false.

The symbol '→' represents implication, which is a logical connective that indicates that the truth of the first proposition (antecedent) implies the truth of the second proposition (consequent). For example, if p is "It is raining" and q is "The ground is wet", then the implication p→q means "If it is raining, then the ground is wet."

The set that contains those elements that are either in A or in B, or in both

The set containing those elements that are in both A and B

The set of all elements in the universal set that are not in set A

The set B

The set of all possible output values of the function

The set of all bit strings of length 3 or greater

The set of all images of elements in S under the function f

The output value produced by the function when applied to a specific input element.

It is either below freezing or it is snowing, but it is not snowing if it is below freezing.

It is below freezing or snowing, but it is not snowing only if it is below freezing.

If you don’t miss the final examination then you pass the course, and conversely.

A compound proposition is a tautology if it is always true, no matter what the truth values of the propositions that occur in it.

The propositions p and q are logically equivalent if 𝑝𝑝 ↔ 𝑞𝑞 is a tautology.

¬(𝑝𝑝 ∨ 𝑞𝑞) ↔ (¬𝑝𝑝 ∧ ¬𝑞𝑞)

This equivalence states that the negation of the disjunction of two propositions is equivalent to the conjunction of their negations.

(𝑝𝑝 → 𝑞𝑞) ↔ (¬𝑞𝑞 ∨ 𝑝𝑝)

A proposition is a contingency if it is neither a tautology nor a contradiction.

The distributive law of logic states that the logical conjunction of a proposition with a logical disjunction is equivalent to the logical disjunction of the logical conjunctions of the proposition with each disjunct.

A proposition is a statement that has a truth value, either true or false.

The negation of a proposition is a proposition that has the opposite truth value.

A proposition is a contradiction if it is always false regardless of the truth values of its variables.

The conditional statement "𝑝𝑝 → 𝑞𝑞" is logically equivalent to "¬𝑝𝑝 ∨ 𝑞𝑞".

The statement "∃𝑥𝑥¬𝑃𝑃(𝑥𝑥)" means "There is a student who does not spend less than three hours every weekday in class."

The statement "∃𝑦𝑦∀𝑥𝑥𝑃𝑃(𝑥𝑥, 𝑦𝑦)" means "There is a computer course at your school taken by all students in your class."

A function is injective (or one-to-one) if each element in the codomain is mapped to by at most one element in the domain.

A function is surjective (or onto) if every element in the codomain is mapped to by at least one element in the domain.

A function is bijective (or one-to-one correspondence) if it is both injective and surjective.

Given functions f and g, the composition g o f is a function where g(f(x)) is calculated for every input x.

The range of a function is the set of all possible outputs (values) that the function can produce.

The operation denoted by ↳ is called floor and it returns the greatest integer less than or equal to the input.

The operation denoted by ↱ is called ceiling and it returns the least integer greater than or equal to the input.

A function is one-to-one (injective) if no two distinct inputs produce the same output. In other words, if f(x) = f(y) then x = y always.

This statement asserts that for every element x in the universe, if x can speak Kazakh (P(x)), then x does not know Delphi (¬Q(x)). In other words, no student at your university can speak Kazakh and know Delphi.

The sentence means that there is a student x in the university who satisfies both conditions: x can speak Kazakh (P(x)) and x does not know Delphi (¬Q(x)). So, there exists a student who can speak Kazakh but is not a Delphi programmer.

This statement means that there exists an element x in the universe of discourse (integers) such that for every element y in the universe of discourse, the statement 'x + y = x * y' is true. In other words, there is an integer x that satisfies the equation with any integer y.

This statement asserts that for every integer x, there exists an integer y such that the equation 'x + 3y = 3x - y' is true. So, no matter what integer you choose for x, there will always be a corresponding integer y that satisfies the equation.

This statement means that for every integer x, there exists an integer y such that 'x + y = x * y' is false. In other words, for every integer x, you can find at least one integer y that does not satisfy the given equation.

This statement means that there exists an integer n such that n2 = 8. In other words, there is an integer whose square is 8.

This statement says that for every real number x, there exists a real number y such that x is equal to the square of y. In other words, every real number has a square root.

This statement says that for every real number x, there exists a real number y such that x multiplied by y equals 4. In other words, every real number can be multiplied by some other real number to equal 4.

The statement is true. You can check it by plugging in the values for x and y. For example, if x = 1 and y = 2, it gives us 1 + 3(2) = 3(1) - 2 which simplifies to 7 = 1. This contradicts the given information.

This statement asserts that for every natural number n, the square of n is always greater than or equal to 1.