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The statement (x implies y) is defined to be the statement ((not x) or y). Intuitively, this makes sense: since x implies y, it should not happen that both x and (not y) hold at the same time. Now we prove: if x implies y and y implies z, then x implies z. If (not x) holds, then we are done, since then ((not x) or z) must hold.
- Conditional Statement
- Biconditional Statement
- Translating Statements and Symbolic Logic
- Converse, Inverse, and Contrapositive
- Truth Values and Truth Tables
- Logical Implication – Lesson & Examples
Here are a few examples of conditional statements: “If it is sunny, then we will go to the beach.” “If the sky is clear, then we will be able to see the stars.” “Studying for the test is a sufficient condition for passing the class.” Here’s a typical list of ways we can express a logical implication: 1. If p, then q 2. If p, q 3. p is sufficient fo...
Now, another necessary type of implication is called a biconditional statement. A biconditional statement, sometimes referred to as a bi-implication, may take one the following forms: 1. P if and only if q 2. P is necessary and sufficient for q 3. If p then q, and conversely 4. P iff q, where “iff” stands for “if and only if” And the biconditional ...
Additionally, we will discover six different types of sentences in propositional logic, and we will learn how to translate from English to symbols and vice versa with ease.
Furthermore, we will learn how to take conditional statements and find new compound statements in the converse, inverse, and contrapositive form. For example, let’s suppose we have the proposition: “If the card is a club, then it is black,” has a very different truth value than “if the card is black, then it is a club.” The first conditional statem...
And being able to verify the truth value of conditional statements and its inverse, converse, and the contrapositive is going to be an essential part of our analysis. Consider the implication: if n is an odd integer, then 5n+1 is even. Write the converse, inverse, contrapositive, and biconditional statements. 1. Converse: if 5n+1 is even, then n is...
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any number y from the interval [x− 1,x+ 1], this number y will satisfy y≥ −x. Do you think this statement is true or false? Now suppose xis fixed. Consider the statement about x: (2) ∀y∈ [x−1,x+1], y≥ −x. Plot xon the real line, plot the interval [x−1,x+1], and find a simple condition on x(in the form of inequalities) which ...
Jul 7, 2021 · The statement \(p\) in an implication \(p \Rightarrow q\) is called its hypothesis, premise, or antecedent, and \(q\) the conclusion or consequence.. Implications come in many disguised forms.
Jul 16, 2021 · Certainly, y can be true when x is false; but x can't be true when y is false. In this case, we say that x implies y. Consider the truth table of \(p \to q\), Table 3.1.7. If p implies q, then the third case can be ruled out, since it is the case that makes a conditional proposition false.
Want to prove that X implies Y. Assume that X is true. Show that this assumption leads to the conclusion that Y must be true. Since this rules out the possibility that X could be true and Y could be false at the same time, it rules out the possibility that "X implies Y" could ever be false. Direct Proof 3
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∀x (P(x) ∨Q(x)) and ∀x(¬Q(x) ∨S(x)) implies ∀x(P(x) ∨S(x)) ∀x (R(x) →¬S(x) is equivalent to ∀x(¬S(x)∨¬R(x)) Therefore ∀x(P(x) ∨¬R(x)) Since ∃x ¬P(x) is true. Thus ¬P(a) for some a in the domain. Since P(a) ∨¬R(a) must be true. Conclusion ¬R(a) is true and so ∃x ¬R(x) is true
