We first designate a symbol to stand for the set itself, usually a capital letter like A or S or some such. Then, we use curly braces to enclose some representation of the elements of the set, as follows:. How do we represent the elements inside the braces? There are two ways. The first and best is simply to list them.
For example, if A is a set of colors, we could write it down this way:. Sometimes, however, listing the elements is not convenient or even possible. Much of the power of set theory arises from the fact that we can form sets whose elements are other sets.
For example, if A , B , and C are sets of colors, we could form a set of sets of colors. Notice that the sets A , B , and C have some elements in common. We consider every set to be a subset of itself. Funny thing to do, really, but it makes sense, sort of.
At least, it matches the definition of subset. Also, there is one set that is a subset of every set, namely the empty set—the set with no elements. This is often denoted by a circle with a line through it, or a pair of braces with nothing between them. The union of two sets A and B is the set containing all the elements that are in either A or B. Thus, if A and B are the two sets of colors above, then we have,.
The intersection of two sets is the set containing only elements that are in both. For example, the intersection of A and C would be denoted as follows:. We will begin where every child begins—with counting. Notice the set notation. This is critical, and provides us with an important characterization:. Thus, the counting numbers—one, two, seventy-three, a million, and so on—are elements of the set of natural numbers.
The set of natural numbers has some properties that should be noted. First, of course, is that this set is ordered. This may seem so obvious as to be beneath our notice, but we will find as we start to really learn mathematics as opposed to just memorizing procedures that such niceties can sometimes take on surprising significance.
In fact we can do better with the natural numbers than saying merely that they are ordered. They have a property that we call being well-ordered:. Notice that there is not always a largest element; the set of all even numbers, the set of all multiples of 5, and the set of all the natural numbers greater than 37 are examples of sets that have no greatest element. This business of not having a largest element is something every child notices at some point and then experiences her or his first brush with the idea of infinity.
We know that there isn't a largest natural number because, intuitively at least, we know the following principle which is sometimes called the Archimedian principle :. Another way of saying this is that the natural numbers are closed under addition. That is, take any two natural numbers and add them, and you get another natural number.
Addition can't take you outside of the set. Notice that the natural numbers are also closed under multiplication which makes sense, since multiplication is just repeated addition. The natural numbers are the only numbers we need for one of the most imporant results in classical mathematics, which comes down to us from antiquity it is found in Euclid's Elements.
This result is called the Fundamental Theorem of Arithmetic , which every numerate person should know. Students often ask why zero isn't included in the set of natural numbers. Well, sometimes it is. Many texts describe a set called the whole numbers , by which is meant the natural numbers with zero included. This terminology isn't used much, however, and mathematicians will usually make clear whether they intend to include zero when discussing or using the natural numbers.
It is well to separate zero conceptually from the natural numbers because it is really a very different kind of thing. Its historical development is quite different, too. This significant difference is ignored by those who claim that times are dominant, so three times as much is actually three times as much.
X times more means additions to what you already own! This is also one hundred percent more or double or double.
Bigger than. Aman is born a child, grows to maturity and then enters old age as he deteriorates and returns to a state of fragility. A is twice the size of B you seem to be talking about numbers.
There are twice as many A than B, as if I were counting things. If I have four brothers and two sisters, I can say I have twice as many auxiliary brothers. Aggregate of two or more numbers, measures, quantities or information, as determined or determined by the thematic addition procedure: the sum of 6 and 8 is The answer to dividing one number by another.
Answer dated September 12, But a number twice as high as another would be three times as high as the other number. Multiplication-product, multiply, multiplied by, times. Subsequently, question is, does or mean add or multiply?
The added assumptions are: you can only add if the two events are disjoint. Is is what is known as a state of being verb. State of being verbs do not express any specific activity or action but instead describe existence. The most common state of being verb is to be, along with its conjugations is, am, are, was, were, being, been. In Algebra a term is either a single number or variable, or numbers and variables multiplied together.
In mathematics , we sometimes use it to mean multiplication, particularly with computers. The addition is taking two or more numbers and adding them together, that is, it is the total sum of 2 or more numbers. Mathematics is the study of numbers, shapes and patterns.
Numbers: how things can be counted. Example: 4! The factorial function symbol:! You use no more than or not more than when you want to emphasize how small a number or amount is. He was a kid really, not more than eighteen or nineteen. Synonyms : sectionalisation, section, class, naval division , part, partitioning, air division , partition, variance, sectionalization, segmentation. So if you subtract the smaller value from the larger value, you will find the difference, or how many more one quantity has than another.
In math " more " means a higher amount of something than something or someone else.
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