Mar 16, 2016 · The first statement, directly phrased, would sound like "For all x, P of x and Q of x if and only if for all x, P of x, and for all x, Q of x." Some minor additions and modifications makes it a little more understandable. "For all x, P and Q are true for x if and only if P is true for all x and Q is true for all x."
Sep 17, 2019 · 9. Mutually Exclusive Events. Two or more events are said to be mutually exclusive if the occurrence of any one of them means the others will not occur (That is, we cannot have 2 or more such events occurring at the same time). For example, if we throw a 6-sided die, the events "4" and "5" are mutually exclusive.
Furthermore, let n be a natural number, and suppose P(m) is true for all natural numbers m less than n + 1. Then if P(n + 1) is false n + 1 is in S, thus being a minimal element in S, a contradiction. Thus P(n + 1) is true. Therefore, by the complete induction principle, P(n) holds for all natural numbers n; so S is empty, a contradiction. Q.E.D.
What does it mean to say, "p is both necessary and sufficient for q." If p, then q is true; and conversely, If q, then p is true. That is, p if and only if q. Problem 13. Which of these if and only if statements is true. Explain. a) A number is divisible by 6 if and only if it is divisible by both 2 and 3. True.
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Minimax Solutions - Example 3.4.1 It should be noted that if Player 1 chose p > 1 4, then @R1(p;q) @q > 0 and Player 2 minimises Player 1’s reward by choosing q = 0, i.e.
∼ q → ∼ p. B. p only if q. C. p is a sufficient condition for q. D. q is a necessary condition for p. Let's analyze the question by looking at the truth table of the implication. From the truth table its clear that p → q is equivalent to ∼ q → ∼ p.
The inverse is "If ~p then ~q." Symbolically, the inverse of p q is ~p ~q. A conditional statement is not logically equivalent to its inverse. Only if : p only if q means "if not q then not p, " or equivalently, "if p then q." Biconditional (iff): The biconditional of p and q is "p if, and only if, q" and is denoted p q.
4. Let p and q be the propositions. p : I bought a lottery ticket this week. q : I won the million dollar jackpot on Friday. Expression each of these propositions as an English sentence. e) p ↔ q. I bought a lottery ticket this week if and only if I won the million dollar jackpot on Friday. f) ¬p → ¬q